Grades 11 / 12 Tests and Answer Keys

Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color? Express your answer as a simple fraction in lowest terms.

  (2 pts) 1.

Find the exact value of

Express your answer as a simple fraction in lowest terms.

  (3 pts) 2.

What is the exact value of $x+y$ for real numbers $x$ and $y$ satisfying the following equation?

  (3 pts) 3.

The real numbers $a$, $b$, and $c$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $a{x}^2+bx+c$ has exactly one root. What is this root?

  (3 pts) 4.

In a triangle $\mathrm{△}ABC$, the angles satisfy the ratio $\mathrm{\angle }A:\mathrm{\angle }B:\mathrm{\angle }C=1:2:3$. If the longest side of $\mathrm{△}ABC$ is 10 units, what is the perimeter of the triangle $\mathrm{△}ABC$?

  (3 pts) 5.

Line $L_1$ has the equation $3x-2y=1$ and passes through the point $A(-1,-2)$. Line $L_2$ is given by $y=1$ and intersects $L_1$ at the point $B$. Line $L_3$, which has a positive slope, passes through point $A$ and intersects $L_2$ at the point $C$. The area of $\mathrm{△}ABC$ is 3. What is the slope of $L_3$?

  (3 pts) 6.

Let $f_1(x)=10x-1$ and $f_n(x)=f_1\left(f_{n-1}(x)\right)$ for integers $n\ge 2$. What is the sum of the digits of $f_{2025}(1)$?

  (3 pts) 7.

Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color? Express your answer as a simple fraction in lowest terms.

  (2 pts) 1. $13/17$

Find the exact value of

Express your answer as a simple fraction in lowest terms.

  (3 pts) 2. $1/16$

What is the exact value of $x+y$ for real numbers $x$ and $y$ satisfying the following equation?

  (3 pts) 3. 2

The real numbers $a$, $b$, and $c$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $a{x}^2+bx+c$ has exactly one root. What is this root?

  (3 pts) 4. $-2+\sqrt{3}$

In a triangle $\mathrm{△}ABC$, the angles satisfy the ratio $\mathrm{\angle }A:\mathrm{\angle }B:\mathrm{\angle }C=1:2:3$. If the longest side of $\mathrm{△}ABC$ is 10 units, what is the perimeter of the triangle $\mathrm{△}ABC$?

  (3 pts) 5. $15+5\sqrt{3}$

Line $L_1$ has the equation $3x-2y=1$ and passes through the point $A(-1,-2)$. Line $L_2$ is given by $y=1$ and intersects $L_1$ at the point $B$. Line $L_3$, which has a positive slope, passes through point $A$ and intersects $L_2$ at the point $C$. The area of $\mathrm{△}ABC$ is 3. What is the slope of $L_3$?

  (3 pts) 6. $3/4$

Let $f_1(x)=10x-1$ and $f_n(x)=f_1\left(f_{n-1}(x)\right)$ for integers $n\ge 2$. What is the sum of the digits of $f_{2025}(1)$?

  (3 pts) 7. $16,201$

Three cards are chosen at random from a standard 52-card deck. What is the probability that they are not all the same color? Express your answer as a simple fraction in lowest terms.

  (2 pts) 1. $13/17$

Solution: Find the probability they are all same color, then subtract that from 1. There are 26 cards of each color, so 3 of them can be selected in $\left(\genfrac{}{}{0pt}{}{26}{3}\right)$ ways, and there are 2 colors. So the answer is $1-{\displaystyle \frac{2\left(\genfrac{}{}{0pt}{}{26}{3}\right)}{\left(\genfrac{}{}{0pt}{}{52}{3}\right)}}={\displaystyle \frac{13}{17}}$.

Find the exact value of

Express your answer as a simple fraction in lowest terms.

  (3 pts) 2. $1/16$

Solution: ${\displaystyle \frac{2^{-2024}(1+2^2)}{2^{-2020}(1+2^2)}}=2^{-4}={\displaystyle \frac{1}{16}}$.

What is the exact value of $x+y$ for real numbers $x$ and $y$ satisfying the following equation?

  (3 pts) 3. 2

Solution: The equation can be expressed as ${(x-5)}^2+{(y+3)}^2=0$. Then $x=5$, $y=-3$, and $x+y=2$.

The real numbers $a$, $b$, and $c$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $a{x}^2+bx+c$ has exactly one root. What is this root?

  (3 pts) 4. $-2+\sqrt{3}$

Solution: Let $a=b+k$ and $c=b-k$ for some $k>0$. Since the quadratic has exactly one root, $b^2-4ac=0\Rightarrow b^2-4(b+k)(b-k)=-3b^2+4k^2=0\Rightarrow k={\displaystyle \frac{\sqrt{3}}{2}}b$. Then the root is $-\frac{b}{2a}=-{\displaystyle \frac{b}{2(b+\frac{\sqrt{3}}{2}b)}}={\displaystyle \frac{-1}{2+\sqrt{3}}}=\sqrt{3}-2$.

In a triangle $\mathrm{△}ABC$, the angles satisfy the ratio $\mathrm{\angle }A:\mathrm{\angle }B:\mathrm{\angle }C=1:2:3$. If the longest side of $\mathrm{△}ABC$ is 10 units, what is the perimeter of the triangle $\mathrm{△}ABC$?

  (3 pts) 5. $15+5\sqrt{3}$

Solution: Let $\mathrm{\angle }A=x$, $\mathrm{\angle }B=2x$, and $\mathrm{\angle }C=3x$. Then $x+2x+3x={180}^{\circ }\Rightarrow x={30}^{\circ }$. So the angles are $\mathrm{\angle }A={30}^{\circ }$, $\mathrm{\angle }B={60}^{\circ }$, and $\mathrm{\angle }C={90}^{\circ }$, and $\mathrm{△}ABC$ is a right triangle. Then, the side lengths are 5, $5\sqrt{3}$, and 10, and the perimeter is $15+5\sqrt{3}$.

Line $L_1$ has the equation $3x-2y=1$ and passes through the point $A(-1,-2)$. Line $L_2$ is given by $y=1$ and intersects $L_1$ at the point $B$. Line $L_3$, which has a positive slope, passes through point $A$ and intersects $L_2$ at the point $C$. The area of $\mathrm{△}ABC$ is 3. What is the slope of $L_3$?

  (3 pts) 6. $3/4$

Solution: The points $A$, $B$, and $C$ form a triangle. The distance from the point $A$ to $L_2$ is 3, which is the height of the triangle, and the length of the line segment between $B$ and $C$ should be 2. Since $L_3$ has a positive slope, the point $C$ is at the point $(3,1)$. Then the slope of $L_3$ is $d\frac{3}{4}$.

Let $f_1(x)=10x-1$ and $f_n(x)=f_1\left(f_{n-1}(x)\right)$ for integers $n\ge 2$. What is the sum of the digits of $f_{2025}(1)$?

  (3 pts) 7. $16,201$

Solution: $f_2(x)=10(10x-1)-1={10}^{2}x-10-1$, $f_3(x)={10}^3-{10}^2-10-1$, $\mathrm{\dots }$, $f_{2025}(x)={10}^{2025}x-{10}^{2024}-\mathrm{\cdots }-10-1$. Then $f_{2025}(1)=8888\mathrm{\cdots }89$ with 2024 eights and one nine. Thus, the sum of the digits is $8\cdot 2024+9=16,201$.

Two rectangles have the same perimeter. In the first rectangle the ratio of the long edge to the short edge is $2:1$ and in the second rectangle these edges are in the ratio of $3:2$. Then the ratio of the area of the first rectangle to the area of the second rectangle is

(a) $2:3$   (b) $25:6$   (c) $25:27$   (d) $27:2$   (e) not determined

  (2 pts) 1.

How many pairs of integers $(x,y)$ satisfy the equation $xy-2x-3y+1=0$?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) infinitely many

  (3 pts) 2.

Given square $ABCD$ with side length 4, points $E$, $F$, $G$, and $H$ are taken on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AE=BF=CG=DH$. Determine the length of $AE$ that minimizes the perimeter of quadrilateral $EFGH$.

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) $5$

  (3 pts) 3.

Let $f$ be the function such that $f(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$ satisfying $f(-1)=0$ and $f(x)-f(x-1)=2x(x+1)(2x+1)$ for any real number $x$. What is the value of $2025a+13b+c$?

(a) $2025$    (b) $2042$    (c) $2082$    (d) $2100$    (e) $2185$

  (3 pts) 4.

Chris’s garden is twice as large as Sarah’s garden and three times as large as James’s. James waters plants half as quickly as Sarah and one-third as quickly as Chris. If they all start watering their gardens at the same time, who will finish watering first?

(a) Chris   (b) Sarah   (c) James and Chris tie for first
(d) Sarah and James tie for first   (e) None of these

  (3 pts) 5.

Peter has $250$ students enrolled in his school club. He decides to form groups so that each group has a different number of students. The maximum number of groups he can form is:

(a) $20$    (b) $21$    (c) $22$    (d) $23$    (e) none of these

  (3 pts) 6.

For how many integers $m\ge 0$ is $m^2+7$ a perfect square?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) no integer is satisfied

  (3 pts) 7.

Two rectangles have the same perimeter. In the first rectangle the ratio of the long edge to the short edge is $2:1$ and in the second rectangle these edges are in the ratio of $3:2$. Then the ratio of the area of the first rectangle to the area of the second rectangle is

(a) $2:3$   (b) $25:6$   (c) $25:27$   (d) $27:2$   (e) not determined

  (2 pts) 1. (c)

How many pairs of integers $(x,y)$ satisfy the equation $xy-2x-3y+1=0$?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) infinitely many

  (3 pts) 2. (d)

Given square $ABCD$ with side length 4, points $E$, $F$, $G$, and $H$ are taken on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AE=BF=CG=DH$. Determine the length of $AE$ that minimizes the perimeter of quadrilateral $EFGH$.

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) $5$

  (3 pts) 3. (b)

Let $f$ be the function such that $f(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$ satisfying $f(-1)=0$ and $f(x)-f(x-1)=2x(x+1)(2x+1)$ for any real number $x$. What is the value of $2025a+13b+c$?

(a) $2025$    (b) $2042$    (c) $2082$    (d) $2100$    (e) $2185$

  (3 pts) 4. (c)

Chris’s garden is twice as large as Sarah’s garden and three times as large as James’s. James waters plants half as quickly as Sarah and one-third as quickly as Chris. If they all start watering their gardens at the same time, who will finish watering first?

(a) Chris   (b) Sarah   (c) James and Chris tie for first
(d) Sarah and James tie for first   (e) None of these

  (3 pts) 5. (b)

Peter has $250$ students enrolled in his school club. He decides to form groups so that each group has a different number of students. The maximum number of groups he can form is:

(a) $20$    (b) $21$    (c) $22$    (d) $23$    (e) none of these

  (3 pts) 6. (b)

For how many integers $m\ge 0$ is $m^2+7$ a perfect square?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) no integer is satisfied

  (3 pts) 7. (a)

Two rectangles have the same perimeter. In the first rectangle the ratio of the long edge to the short edge is $2:1$ and in the second rectangle these edges are in the ratio of $3:2$. Then the ratio of the area of the first rectangle to the area of the second rectangle is

(a) $2:3$   (b) $25:6$   (c) $25:27$   (d) $27:2$   (e) not determined

  (2 pts) 1. (c)

How many pairs of integers $(x,y)$ satisfy the equation $xy-2x-3y+1=0$?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) infinitely many

  (3 pts) 2. (d)

Given square $ABCD$ with side length 4, points $E$, $F$, $G$, and $H$ are taken on sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $AE=BF=CG=DH$. Determine the length of $AE$ that minimizes the perimeter of quadrilateral $EFGH$.

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) $5$

  (3 pts) 3. (b)

Let $f$ be the function such that $f(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$ satisfying $f(-1)=0$ and $f(x)-f(x-1)=2x(x+1)(2x+1)$ for any real number $x$. What is the value of $2025a+13b+c$? (a) $2025$    (b) $2042$    (c) $2082$    (d) $2100$    (e) $2185$

  (3 pts) 4. (c)

Chris’s garden is twice as large as Sarah’s garden and three times as large as James’s. James waters plants half as quickly as Sarah and one-third as quickly as Chris. If they all start watering their gardens at the same time, who will finish watering first?

(a) Chris   (b) Sarah   (c) James and Chris tie for first
(d) Sarah and James tie for first   (e) None of these

  (3 pts) 5. (b)

The watering rates are:

• •

  James: $r$,

• •

  Sarah: $2r$,

• •

  Chris: $3r$.

The time to water each garden is:

James: $\frac{x}{r}$,  Sarah: $\frac{\frac{3x}{2}}{2r}=\frac{3x}{4r}$,  Chris: $\frac{3x}{3r}=\frac{x}{r}$.

Clearly, , so Sarah will finish watering first.

Peter has $250$ students enrolled in his school club. He decides to form groups so that each group has a different number of students. The maximum number of groups he can form is:

(a) $20$    (b) $21$    (c) $22$    (d) $23$    (e) none of these

  (3 pts) 6. (b)

For how many integers $m\ge 0$ is $m^2+7$ a perfect square?

(a) $1$    (b) $2$    (c) $3$    (d) $4$    (e) no integer is satisfied

  (3 pts) 7. (a)

Alex needs to create a four-digit code for his locker. The digits must be selected from the set $\{0,1,2,3,4,5,6,7,8,9\}$, and they must be distinct. To make it easier to remember, Alex decides to arrange the digits in strictly decreasing order. For instance, the code 9530 follows this rule. How many different four-digit codes can Alex choose?

  (2 pts) 1.

Elsa shopped at a store where each item costs either $1.00 or 49 cents. She spent a total of $39.84 How many items did she purchase?

  (3 pts) 2.

What is the coefficient of the $x^3{y}^7$ term in the expansion of ${(2x+y)}^{10}$ ?

  (3 pts) 3.

If you draw four lines in the plane, they divide the plane into some number of regions. What is the maximum number of regions you can get?

  (3 pts) 4.

What is the smallest positive integer $x$ such that $x^2$ has 2024 as a factor?

  (3 pts) 5.

The arithmetic mean $A$, of any two positive numbers $x$ and $y$ is defined to be $A=\frac{1}{2}(x+y)$ and their geometric mean, $G$ is defined to be $G=\sqrt{xy}$. For two particular values $x$ and $y$, with $x>y$, the ratio $A:G=5:4$. For these values of $x$ and $y$, what is the ratio $x:y$ ?

  (3 pts) 6.

The diagram shows a square ABCD and a right-angled triangle ABE. The length of BC is 4 and the length of BE is 5. What is the area of the shaded region?

  (3 pts) 7.

Alex needs to create a four-digit code for his locker. The digits must be selected from the set $\{0,1,2,3,4,5,6,7,8,9\}$, and they must be distinct. To make it easier to remember, Alex decides to arrange the digits in strictly decreasing order. For instance, the code 9530 follows this rule. How many different four-digit codes can Alex choose?

  (2 pts) 1. 210

Elsa shopped at a store where each item costs either $1.00 or 49 cents. She spent a total of $39.84 How many items did she purchase?

  (3 pts) 2. $32+16=48$

What is the coefficient of the $x^3{y}^7$ term in the expansion of ${(2x+y)}^{10}$ ?

  (3 pts) 3. 960

If you draw four lines in the plane, they divide the plane into some number of regions. What is the maximum number of regions you can get?

  (3 pts) 4. 11

What is the smallest positive integer $x$ such that $x^2$ has 2024 as a factor?

  (3 pts) 5. 1012

The arithmetic mean $A$, of any two positive numbers $x$ and $y$ is defined to be $A=\frac{1}{2}(x+y)$ and their geometric mean, $G$ is defined to be $G=\sqrt{xy}$. For two particular values $x$ and $y$, with $x>y$, the ratio $A:G=5:4$. For these values of $x$ and $y$, what is the ratio $x:y$ ?

  (3 pts) 6. $4:1$

The diagram shows a square ABCD and a right-angled triangle ABE. The length of BC is 4 and the length of BE is 5. What is the area of the shaded region?

  (3 pts) 7. $\frac{48}{5}$ or $9.6$

A box contains 50 balls numbered 1 through 50. Jake draws a ball at random, then Sally draws a ball at random from the remaining 49. If the product of the numbers they draw is even, Sally wins. If the product is odd, Jake wins. Which of the following is true?

1. a.

   Jake is more likely to win.

2. b.

   Jake and Sally are equally likely to win.

3. c.

   Sally is more likely to win.

  (2 pts) 1.

What is the area of the shaded region inside the square?

  (3 pts) 2.

If $a$ is a positive number and $x^4+16=(x^2+ax+4)(x^2-ax+4)$, find $a$.

  (3 pts) 3.

Find $x$ if $x$ is positive and: $x={\displaystyle \frac{1}{\frac{1}{x+3}+\frac{1}{x+12}}}$.

  (3 pts) 4.

Compute $\sqrt{{24}^2+25+24}$

  (3 pts) 5.

Find the length of side $x$ given the information in the picture.

  (3 pts) 6.

Compute ${190}^2-{198}^2$

  (3 pts) 7.

A box contains 50 balls numbered 1 through 50. Jake draws a ball at random, then Sally draws a ball at random from the remaining 49. If the product of the numbers they draw is even, Sally wins. If the product is odd, Jake wins. Which of the following is true?

1. a.

   Jake is more likely to win.

2. b.

   Jake and Sally are equally likely to win.

3. c.

   Sally is more likely to win.

  (2 pts) 1. c

What is the area of the shaded region inside the square?

  (3 pts) 2. 6

If $a$ is a positive number and $x^4+16=(x^2+ax+4)(x^2-ax+4)$, find $a$.

  (3 pts) 3. $2\sqrt{2}$ or $\sqrt{8}$

Find $x$ if $x$ is positive and: $x={\displaystyle \frac{1}{\frac{1}{x+3}+\frac{1}{x+12}}}$.

  (3 pts) 4. 6

Compute $\sqrt{{24}^2+25+24}$

  (3 pts) 5. 25

Find the length of side $x$ given the information in the picture.

  (3 pts) 6. 5

Compute ${190}^2-{198}^2$

  (3 pts) 7. 379

A box contains 50 balls numbered 1 through 50. Jake draws a ball at random, then Sally draws a ball at random from the remaining 49. If the product of the numbers they draw is even, Sally wins. If the product is odd, Jake wins. Which of the following is true?

1. a.

   Jake is more likely to win.

2. b.

   Jake and Sally are equally likely to win.

3. c.

   Sally is more likely to win.

Solution: Jake only wins if both numbers are odd. The chances of that are:

What is the area of the shaded region inside the square?

Solution: The shaded area is: $9-2(\frac{1}{2}(3)(1))=6$

If $a$ is a positive number and $x^4+16=(x^2+ax+4)(x^2-ax+4)$, find $a$.

Solution: Note that

So we need $a>0$ with $a^2=8$, hence $a=2\sqrt{2}$.

Find $x$ if $x$ is positive and: $x={\displaystyle \frac{1}{\frac{1}{x+3}+\frac{1}{x+12}}}$.
Solution:

Since $x$ is positive, $x=6$.

Compute $\sqrt{{24}^2+25+24}$
Solution:

Find the length of side $x$ given the information in the picture.

Solution: Using the Pythagorean Theorem we have:

Compute ${190}^2-{198}^2$
Solution:

A rectangular piece of plywood is 3.6 feet wide and 7.2 feet long. Find the area of the piece of plywood. Round your answer to one decimal place. Express your answer in square feet.

  (20 pts) 1.

A used car dealer has six cars to sell. Two of the cars are red, and the other four cars are white. The dealer wishes to park all six of the cars in a row in front of the showroom building. The row of cars will be along the street, and all six cars will face the street. The dealer has enough space to park six cars along the street, but no more than six cars. The two red cars must be parked adjacent to each other. How many arrangements of the six cars are possible?

  (20 pts) 2.

A large bowl contains 22 balls. All of the balls are of the same size and shape, and all are made of the same material. Of the 22 balls, 9 are red, 7 are white, and 6 are blue. Suppose that we randomly select two of the 22 balls and place both of the selected balls together in a different bowl. What is the probability that neither of the selected balls is red?

  (20 pts) 3.

Mrs. Noriega is designing a box. The box is to have six sides. Each of the six sides is to be a rectangle. The dimensions of the box, in inches, are to be $x$ by $12-2x$ by $9-2x$. Here $x$ is a number which Mrs. Noriega has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three of the dimensions of the box must be positive real numbers.

  (20 pts) 4.

The cost of a ticket for a certain baseball game was $25. Children under the age of 18 received a discount, however. The cost of a “youth” ticket was $17. The team sold 748 tickets for the game. The total sales revenue for these 748 tickets was $17,532. How many youth tickets did the team sell?

  (20 pts) 5.

What is the area of the region in the $xy$-plane whose points $(x,y)$ satisfy the following inequality?

  (20 pts) 6.

Mrs. Ramaphosa drives ten miles to work each day. She leaves for work at the same time every morning and arrives at work at exactly 8:00 A.M. Her average travel speed each morning is 40 miles per hour. One morning, however, the traffic is congested. After Mrs. Ramaphosa has traveled exactly two miles, she determines that her average speed up to that point has been only 24 miles per hour. If Mrs. Ramaphosa is to arrive at work at exactly 8:00 A.M., what must her average speed for the remaining 8 miles of the trip be? Hint: Find the required average speed for the last 8 miles of the trip only.

  (20 pts) 7.

Suppose that two sides of an isosceles triangle have length 3, and let $x$ equal the length of the third side of the triangle. What value of $x$ will produce a triangle with the largest possible area? Round your answer to two decimal places.

  (20 pts) 8.

A disk is a circle together with the points inside the circle. Suppose that disks of equal radii are arranged in a regular pattern throughout the plane in such a way that each disk touches and is tangent to six other disks. What percentage of the plane is covered by the disks? Round your answer to two decimal places. For example, your answer might be 72.34%, 47.83%, or something similar to these percentages.

  (20 pts) 9.

Four solid spheres lie on the top of a table. Each sphere is tangent to each of the other three spheres. Three of the four spheres have radius 4 cm, and the other sphere has a radius of less than 4 cm. What is the radius of the smaller sphere? Round your answer to two decimal places.

  (20 pts) 10.

A rectangular piece of plywood is 3.6 feet wide and 7.2 feet long. Find the area of the piece of plywood. Round your answer to one decimal place. Express your answer in square feet.

  (20 pts) 1. 25.9 sq ft

A used car dealer has six cars to sell. Two of the cars are red, and the other four cars are white. The dealer wishes to park all six of the cars in a row in front of the showroom building. The row of cars will be along the street, and all six cars will face the street. The dealer has enough space to park six cars along the street, but no more than six cars. The two red cars must be parked adjacent to each other. How many arrangements of the six cars are possible?

  (20 pts) 2. 240

A large bowl contains 22 balls. All of the balls are of the same size and shape, and all are made of the same material. Of the 22 balls, 9 are red, 7 are white, and 6 are blue. Suppose that we randomly select two of the 22 balls and place both of the selected balls together in a different bowl. What is the probability that neither of the selected balls is red?

  (20 pts) 3. 0.338 or 0.34 or 33.8% or 34% or $\frac{26}{77}$, but not 0.3 or $\frac{3}{10}$

Mrs. Noriega is designing a box. The box is to have six sides. Each of the six sides is to be a rectangle. The dimensions of the box, in inches, are to be $x$ by $12-2x$ by $9-2x$. Here $x$ is a number which Mrs. Noriega has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three of the dimensions of the box must be positive real numbers.

  (20 pts) 4. 1.70

The cost of a ticket for a certain baseball game was $25. Children under the age of 18 received a discount, however. The cost of a “youth” ticket was $17. The team sold 748 tickets for the game. The total sales revenue for these 748 tickets was $17,532. How many youth tickets did the team sell?

  (20 pts) 5. 146

What is the area of the region in the $xy$-plane whose points $(x,y)$ satisfy the following inequality?

  (20 pts) 6. 3

Mrs. Ramaphosa drives ten miles to work each day. She leaves for work at the same time every morning and arrives at work at exactly 8:00 A.M. Her average travel speed each morning is 40 miles per hour. One morning, however, the traffic is congested. After Mrs. Ramaphosa has traveled exactly two miles, she determines that her average speed up to that point has been only 24 miles per hour. If Mrs. Ramaphosa is to arrive at work at exactly 8:00 A.M., what must her average speed for the remaining 8 miles of the trip be? Hint: Find the required average speed for the last 8 miles of the trip only.

  (20 pts) 7. 48 mph

Suppose that two sides of an isosceles triangle have length 3, and let $x$ equal the length of the third side of the triangle. What value of $x$ will produce a triangle with the largest possible area? Round your answer to two decimal places.

  (20 pts) 8. 4.24

A disk is a circle together with the points inside the circle. Suppose that disks of equal radii are arranged in a regular pattern throughout the plane in such a way that each disk touches and is tangent to six other disks. What percentage of the plane is covered by the disks? Round your answer to two decimal places. For example, your answer might be 72.34%, 47.83%, or something similar to these percentages.

  (20 pts) 9. 90.69%

Four solid spheres lie on the top of a table. Each sphere is tangent to each of the other three spheres. Three of the four spheres have radius 4 cm, and the other sphere has a radius of less than 4 cm. What is the radius of the smaller sphere? Round your answer to two decimal places.

  (20 pts) 10. 1.33cm

A rectangular piece of plywood is 3.6 feet wide and 7.2 feet long. Find the area of the piece of plywood. Round your answer to one decimal place. Express your answer in square feet.

Solution: $3.6\cdot 7.2=25.92\approx 25.9$

A used car dealer has six cars to sell. Two of the cars are red, and the other four cars are white. The dealer wishes to park all six of the cars in a row in front of the showroom building. The row of cars will be along the street, and all six cars will face the street. The dealer has enough space to park six cars along the street, but no more than six cars. The two red cars must be parked adjacent to each other. How many arrangements of the six cars are possible?

Solution: Think of the six parking places along the street. We may number these parking places from 1 to 6. The leftmost of the red cars can be in spots 1, 2, 3, 4, or 5. There are 2 ways of arranging the two red cars in this spot and its neighbor to the right. There are $4\cdot 3\cdot 2\cdot 1=24$ ways to arrange the four white cars in the four slots not occupied by red cars. The number of ways of arranging the six cars is thus $5\cdot 2\cdot 24=240$.

A large bowl contains 22 balls. All of the balls are of the same size and shape, and all are made of the same material. Of the 22 balls, 9 are red, 7 are white, and 6 are blue. Suppose that we randomly select two of the 22 balls and place both of the selected balls together in a different bowl. What is the probability that neither of the selected balls is red?

Solution: Imagine that we select the two balls one at a time. The probability that the first ball is not red is $\frac{13}{22}$. The probability that the second ball is not red, given that the first ball was not red is $\frac{12}{21}$. The probability that neither ball is red is $\frac{13}{22}\cdot 1221=\frac{13\cdot 2}{11\cdot 7}=\frac{27}{77}\approx 0.338$.

Mrs. Noriega is designing a box. The box is to have six sides. Each of the six sides is to be a rectangle. The dimensions of the box, in inches, are to be $x$ by $12-2x$ by $9-2x$. Here $x$ is a number which Mrs. Noriega has not yet determined. Find the value of $x$ which will result in the box with the largest possible volume. Round your answer to two decimal places. Hint: All three of the dimensions of the box must be positive real numbers.

Solution: We require $x>0$, $12-2x>0$, and $9-2x>0$. Thus and , so that . The volume of the box will be $V(x)=x(12-2x)(9-2x)$. Graphing $V(x)$ and zooming in, we find that the maximum value of $V$ occurs when $x=1.70$.

The cost of a ticket for a certain baseball game was $25. Children under the age of 18 received a discount, however. The cost of a “youth” ticket was $17. The team sold 748 tickets for the game. The total sales revenue for these 748 tickets was $17,532. How many youth tickets did the team sell?

Solution: Let $x$ be the number of adult tickets sold and $y$ be the number of youth tickets sold. Then

By (3.1), $x=748-y$. Plugging this into (3.2), we find that

What is the area of the region in the $xy$-plane whose points $(x,y)$ satisfy the following inequality?

Solution: In quadrant I, $x,y\ge 0$ and the summands are already positive. Therefore, $2x+2y\le 2$, which is the triangle with vertices $(0,1)$, $(1,0)$, and the origin. This has area $\frac{1}{2}$.

In quadrant II: when $x+y\ge 0$, the inequality becomes $-x+y+x+y\le 2$, so that $y\le 1$. When $x+y\le 0$, the inequality becomes $-x+y-x-y\le 2$ so that $-2x\le 2$ and $x\ge -1$. This describes the square with vertices $(-1,0)$, $(-1,1)$, $(0,1)$, and the origin, which has area 1.

In quadrant III, the summands are all negative, so that $-2x-2y\le 2$ and $x+y\ge -1$. This is the triangle with vertices $(-1,0)$, $(0,-1)$, and the origin, which has area $\frac{1}{2}$.

In quadrant IV: when $x+y\ge 0$, the inequality becomes $x-y+x+y\le 2$, so that $x\le 1$. When $x+y\le 0$, the inequality becomes $x-y-x-y\le 2$, so that $-2y\le 2$ and $y\ge -1$. This describes the square with vertices $(1,0)$, $(1,-1)$, $(0,-1)$, and the origin, which has area 1.

The total area of the original region is $\frac{1}{2}+1+\frac{1}{2}+1=3$.

Mrs. Ramaphosa drives ten miles to work each day. She leaves for work at the same time every morning and arrives at work at exactly 8:00 A.M. Her average travel speed each morning is 40 miles per hour. One morning, however, the traffic is congested. After Mrs. Ramaphosa has traveled exactly two miles, she determines that her average speed up to that point has been only 24 miles per hour. If Mrs. Ramaphosa is to arrive at work at exactly 8:00 A.M., what must her average speed for the remaining 8 miles of the trip be? Hint: Find the required average speed for the last 8 miles of the trip only.

Solution: If Mrs. Ramaphosa’s average speed is 40 miles per hour, then it takes her

to drive to work, and she leaves for work at 7:45 A.M. each day. For the morning in question, she has traveled 2 miles after

or 7:50 A.M. If Mrs. Ramaphosa is to arrive at work at exactly 8:00 A.M., she must cover the remaining 8 miles of her trip to work in exactly 10 minutes. Her average speed for the remainder of the trip must be

Suppose that two sides of an isosceles triangle have length 3, and let $x$ equal the length of the third side of the triangle. What value of $x$ will produce a triangle with the largest possible area? Round your answer to two decimal places.

Solution: By the Pythagorean Theorem, the height of the triangle is $h=\sqrt{9-x^2/4}$, so that the area is $A=\frac{1}{2}x\sqrt{9-x^2/4}$. The area is maximized by maximizing

This happens when $x=\sqrt{18}=3\sqrt{2}\approx 4.24$

A disk is a circle together with the points inside the circle. Suppose that disks of equal radii are arranged in a regular pattern throughout the plane in such a way that each disk touches and is tangent to six other disks. What percentage of the plane is covered by the disks? Round your answer to two decimal places. For example, your answer might be 72.34%, 47.83%, or something similar to these percentages.

Solution: Consider an equilateral triangle whose vertices are the centers of three mutually adjacent disks. We can rearrange the portion covered by the disks to see that this has area $\frac{\pi r^2}{2}$. The triangle has base $2r$ and height $r\sqrt{3}$, so that its area is $r^2\sqrt{3}$. We can cover the plane with these triangles, so that the percentage covered is

Four solid spheres lie on the top of a table. Each sphere is tangent to each of the other three spheres. Three of the four spheres have radius 4 cm, and the other sphere has a radius of less than 4 cm. What is the radius of the smaller sphere? Round your answer to two decimal places.

Solution: The centers of the larger spheres forms an equilateral triangle with side length 8. The smaller sphere is centered at the center of the triangle, so that where it touches the table is $8/\sqrt{3}$ from where each larger sphere touches the table.

Suppose the smaller sphere has radius $r$, so that its center is $4+r$ from the center of a larger sphere. Because the centers are horizontally $8/\sqrt{3}$ apart, and vertically $4-r$ apart, the Pythagorean Theorem tells us that

Recall $i^2=-1$. Simplify ${(1+i)}^8$ as much as possible.

  (20 pts) 1.

What is the average of the three solutions to $x^3+6x^2-8x-16$?

  (20 pts) 2.

Find all positive values for a radius of a circle for which the area of the circle is equal numerically to twice its circumference.

  (20 pts) 3.

Find all solutions to $x-1=\sqrt{2x+1}$.

  (20 pts) 4.

Assume today is Friday. What day of the week will it be in 365 days?

  (20 pts) 5.

How many edges does a tetrahedron have?

  (20 pts) 6.

How many real roots does the equation $\cos x={\log }_{10}x$ have?

  (20 pts) 7.

Sort the integers 1 through 4 randomly, and denote them as $a$, $b$, $c$, and $d$. What is the probability that $ab+cd$ is an even number?

  (20 pts) 8.

A penny is placed flat on a table. What is the maximum number of pennies that can be places around it, flat on the table, with each one tangent to it?

  (20 pts) 9.

In a group of dogs and people, the number of legs was 14 more than twice the number of heads. How many dogs were there? [Assume none of the people or dogs is missing a leg or has an extra.]

  (20 pts) 10.

Recall $i^2=-1$. Simplify ${(1+i)}^8$ as much as possible.

  (20 pts) 1. $16$

What is the average of the three solutions to $x^3+6x^2-8x-16$?

  (20 pts) 2. $-2$

Find all positive values for a radius of a circle for which the area of the circle is equal numerically to twice its circumference.

  (20 pts) 3. $4$

Find all solutions to $x-1=\sqrt{2x+1}$.

  (20 pts) 4. $x=4$

Assume today is Friday. What day of the week will it be in 365 days?

  (20 pts) 5. Saturday

How many edges does a tetrahedron have?

  (20 pts) 6. $6$

How many real roots does the equation $\cos x={\log }_{10}x$ have?

  (20 pts) 7. $3$

Sort the integers 1 through 4 randomly, and denote them as $a$, $b$, $c$, and $d$. What is the probability that $ab+cd$ is an even number?

  (20 pts) 8. ${\displaystyle \frac{2}{3}}$

A penny is placed flat on a table. What is the maximum number of pennies that can be places around it, flat on the table, with each one tangent to it?

  (20 pts) 9. $6$

In a group of dogs and people, the number of legs was 14 more than twice the number of heads. How many dogs were there? [Assume none of the people or dogs is missing a leg or has an extra.]

  (20 pts) 10. $7$

Recall $i^2=-1$. Simplify ${(1+i)}^8$ as much as possible.