Grades 9 / 10 Tests and Answer Keys

Each of the numbers 1, 5, 6, 7, 13, 14, 17, 22, and 26 is placed in a different spot below. The numbers 13 and 17 are placed as shown. Joe calculates the average of the numbers in the first three spots, the average of the numbers in the middle three spots, and the average of the numbers in the last three spots. These three averages are equal. What number is placed in the spot to the right of the 17?

text text text 13 17 text text text text

  (2 pts) 1.

A digital clock shows a time of 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order?

  (3 pts) 2.

Suppose that $p$ and $q$ are two different prime numbers and that $n=p^2{q}^2$ What is the number of possible values of $n$ with ?

  (3 pts) 3.

On Monday, Jill traveled $x$ miles at a constant speed of 90 mph. On Tuesday, she traveled on the same route at a constant speed of 120 mph. Her trip on Tuesday took 16 minutes less than her trip on Monday. What is the value of $x$?

  (3 pts) 4.

A rectangular prism has a volume of 12. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is volume of the new prism?

  (3 pts) 5.

In a survey, 100 students were asked if they liked dogs and if they liked cats. A total of 68 students like dogs. A total of 53 like cats. A total of 6 like neither cats nor dogs. How many of the 100 students like both cats and dogs?

  (3 pts) 6.

On a rectangular table 5 units long and 2 units wide, a ball is rolled from point P at an angle of 45^∘ to PQ and bounces off SR. The ball continues to bounce off the sides at 45^∘ until it reaches S. How many bounces of the ball are required?

  (3 pts) 7.

Each of the numbers 1, 5, 6, 7, 13, 14, 17, 22, and 26 is placed in a different spot below. The numbers 13 and 17 are placed as shown. Joe calculates the average of the numbers in the first three spots, the average of the numbers in the middle three spots, and the average of the numbers in the last three spots. These three averages are equal. What number is placed in the spot to the right of the 17?

text text text 13 17 text text text text

  (2 pts) 1. 7

A digital clock shows a time of 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order?

  (3 pts) 2. 458

Suppose that $p$ and $q$ are two different prime numbers and that $n=p^2{q}^2$ What is the number of possible values of $n$ with ?

  (3 pts) 3. 7

On Monday, Jill traveled $x$ miles at a constant speed of 90 mph. On Tuesday, she traveled on the same route at a constant speed of 120 mph. Her trip on Tuesday took 16 minutes less than her trip on Monday. What is the value of $x$?

  (3 pts) 4. 96

A rectangular prism has a volume of 12. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is volume of the new prism?

  (3 pts) 5. 144

In a survey, 100 students were asked if they liked dogs and if they liked cats. A total of 68 students like dogs. A total of 53 like cats. A total of 6 like neither cats nor dogs. How many of the 100 students like both cats and dogs?

  (3 pts) 6. 27

On a rectangular table 5 units long and 2 units wide, a ball is rolled from point P at an angle of 45^∘ to PQ and bounces off SR. The ball continues to bounce off the sides at 45^∘ until it reaches S. How many bounces of the ball are required?

  (3 pts) 7. 5

What is the sum of two roots of the equation $3+{\displaystyle \frac{1}{x-1}}=x-1$ ?

  (2 pts) 1.

What is the last digit of ${47}^{95}$ ?

  (3 pts) 2.

$x-1$ divides $2k{x}^2+kx+1$ where $k$ is a constant. What is $k$ ?

  (3 pts) 3.

If $-2\le x\le 3$ and $-4\le y\le 5$, then $a\le x^2+y\le b$. What is $a+b$ ?

  (3 pts) 4.

If $(a,a)$ is the point on the line $y=x$ closest to the point $(4,3)$, what is $a$ ?

  (3 pts) 5.

If a sequence $\{a_n\}$ converges to $L$ and $a_{n+1}={\displaystyle \frac{2a_n+3}{4}}$, what is $L$?

  (3 pts) 6.

If $\left||x-3|-3\right|\le 3$, then $a\le x\le b$. What is $a+b$ ?

  (3 pts) 7.

What is the sum of two roots of the equation $3+{\displaystyle \frac{1}{x-1}}=x-1$ ?

  (2 pts) 1. $5$

What is the last digit of ${47}^{95}$ ?

  (3 pts) 2. $3$

$x-1$ divides $2k{x}^2+kx+1$ where $k$ is a constant. What is $k$ ?

  (3 pts) 3. $k=-\frac{1}{3}$

If $-2\le x\le 3$ and $-4\le y\le 5$, then $a\le x^2+y\le b$. What is $a+b$ ?

  (3 pts) 4. $a+b=10$

If $(a,a)$ is the point on the line $y=x$ closest to the point $(4,3)$, what is $a$ ?

  (3 pts) 5. $a=\frac{7}{2}$

If a sequence $\{a_n\}$ converges to $L$ and $a_{n+1}={\displaystyle \frac{2a_n+3}{4}}$, what is $L$?

  (3 pts) 6. $L=\frac{3}{2}$

If $\left||x-3|-3\right|\le 3$, then $a\le x\le b$. What is $a+b$ ?

  (3 pts) 7. $a+b=6$

What is the sum of two roots of the equation $3+{\displaystyle \frac{1}{x-1}}=x-1$ ?

  (2 pts) 1. $5$

Solution: $(x-1)(x-4)=1\Rightarrow x^2-5x+3=0$

What is the last digit of ${47}^{95}$ ?

  (3 pts) 2. $3$

Solution: ${47}^{95}={({47}^4)}^{23}\cdot {47}^3\equiv 3mod10$

$x-1$ divides $2k{x}^2+kx+1$ where $k$ is a constant. What is $k$ ?

  (3 pts) 3. $k=-\frac{1}{3}$

Solution: $2k{x}^2+kx+1=(x-1)(2kx+3k)+3k+1$

If $-2\le x\le 3$ and $-4\le y\le 5$, then $a\le x^2+y\le b$. What is $a+b$ ?

  (3 pts) 4. $a+b=10$

Solution: $0\le x^2\le 9\Rightarrow -4\le x^2+y\le 14$

If $(a,a)$ is the point on the line $y=x$ closest to the point $(4,3)$, what is $a$ ?

  (3 pts) 5. $a=\frac{7}{2}$

Solution: $(a-4,a-3)\cdot (1,1)=0\Rightarrow 2a-7=0$

If a sequence $\{a_n\}$ converges to $L$ and $a_{n+1}={\displaystyle \frac{2a_n+3}{4}}$, what is $L$?

  (3 pts) 6. $L=\frac{3}{2}$

Solution: $L={\displaystyle \frac{2L+3}{4}}\Rightarrow 4L=2L+3$

If $\left||x-3|-3\right|\le 3$, then $a\le x\le b$. What is $a+b$ ?

  (3 pts) 7. $a+b=6$

Solution: $-3\le |x-3|-3\le 3\Rightarrow |x-3|\le 6\Rightarrow -6\le x-3\le 6\Rightarrow -3\le x\le 9$

The $y$-intercept of the line through $(2,a)$ and $(a,-7)$ with slope ${\displaystyle \frac{1}{2}}$ is:

  (2 pts) 1.

If $x^3-x^2+kx+3$ is divisible by $x+1$, then it is also divisible by

(a) $2x-1$   (b) $2x+2$   (c) $x^2-2x+3$   (d) $x^2+2x+1$   (e) $x^2-2x+1$

  (3 pts) 2.

An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?

  (3 pts) 3.

The circumference of a circle is 100 inches. The side of a square inscribed in this circle, expressed in inches, is:

(a) $100\sqrt{2}/\pi$   (b) $50\sqrt{2}$   (c) $100/\pi$   (d) $50\sqrt{2}/\pi$   (e) $25\sqrt{2}/\pi$

  (3 pts) 4.

The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years 2014, 2015, and 2017 were 6, 45, and 117, respectively, then what was the population in 2016?

  (3 pts) 5.

A high school science club plans to rent a minivan for a weekend trip to visit a science museum at a cost of $120. If they invite two nonmembers along, each member can save $10 on his or her share of the cost. How many members are in the science club?

  (3 pts) 6.

The graphs of $y=-|x-a|+b$ and $y=|x-c|+d$ intersect at points $(2,5)$ and $(8,3)$. Find $a+c$.

  (3 pts) 7.

The $y$-intercept of the line through $(2,a)$ and $(a,-7)$ with slope ${\displaystyle \frac{1}{2}}$ is:

  (2 pts) 1. $-5$

If $x^3-x^2+kx+3$ is divisible by $x+1$, then it is also divisible by

(a) $2x-1$   (b) $2x+2$   (c) $x^2-2x+3$   (d) $x^2+2x+1$   (e) $x^2-2x+1$

  (3 pts) 2. (c)

An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?

  (3 pts) 3. $48$

The circumference of a circle is 100 inches. The side of a square inscribed in this circle, expressed in inches, is:

(a) $100\sqrt{2}/\pi$   (b) $50\sqrt{2}$   (c) $100/\pi$   (d) $50\sqrt{2}/\pi$   (e) $25\sqrt{2}/\pi$

  (3 pts) 4. (d)

The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years 2014, 2015, and 2017 were 6, 45, and 117, respectively, then what was the population in 2016?

  (3 pts) 5. $60$

A high school science club plans to rent a minivan for a weekend trip to visit a science museum at a cost of $120. If they invite two nonmembers along, each member can save $10 on his or her share of the cost. How many members are in the science club?

  (3 pts) 6. $4$

The graphs of $y=-|x-a|+b$ and $y=|x-c|+d$ intersect at points $(2,5)$ and $(8,3)$. Find $a+c$.

  (3 pts) 7. $10$

Your car is currently 3 years old and is worth $10,000. Two years ago it was worth $12,400. Assume the car’s value depreciates linearly with time. What will be the value of the car in 5 years?

  (2 pts) 1.

Amtrak’s annual passenger revenue for the years 1990-2010 is given by the equation $R=-40|x-11|+990$ where $R$ is the annual revenue in millions of dollars and $x$ is the number of years since January 1, 1990. In what years was the passenger revenue $790 million?

  (3 pts) 2.

Real numbers $a$ and $b$ satisfy the equations $3^a={81}^{b+2}$ and ${125}^b=5^{a-3}$. What is $ab$?

  (3 pts) 3.

Students from Mrs. Pohl’s class are standing in a circle. They are evenly spaced and consecutively numbered starting with 1. The student with number 3 is standing directly across form the student with number 17. How many students are there in Mrs. Pohl’s class?

  (3 pts) 4.

Alice and Bill are walking in opposite directions along the same route between $A$ and $B$. Alice is going from $A$ to $B$, and Bill from $B$ to $A$. They start at the same time. They pass each other 3 hours later. Alice arrives at $B$ 2.5 hours before Bill arrives at $A$. How many hours does it take for Bill to go from $B$ to $A$?

  (3 pts) 5.

You have two boxes. Each of them has a square base and is half as tall as it is wide. If the larger box is two inches wider than the smaller box, and has a volume 244 in³ greater, what is the width of the smaller box?

  (3 pts) 6.

In the triangle $\mathrm{\Delta }ABC$, the point $D$ lies on side $\overline{BC}$. Also, $AC=3$, $AD=3$, $BD=8$, and $CD=1$. What is AB?

  (3 pts) 7.

Your car is currently 3 years old and is worth $10,000. Two years ago it was worth $12,400. Assume the car’s value depreciates linearly with time. What will be the value of the car in 5 years?

  (2 pts) 1. $4000

Amtrak’s annual passenger revenue for the years 1990-2010 is given by the equation $R=-40|x-11|+990$ where $R$ is the annual revenue in millions of dollars and $x$ is the number of years since January 1, 1990. In what years was the passenger revenue $790 million?

  (3 pts) 2. 1996 and 2006

Real numbers $a$ and $b$ satisfy the equations $3^a={81}^{b+2}$ and ${125}^b=5^{a-3}$. What is $ab$?

  (3 pts) 3. $60$

Students from Mrs. Pohl’s class are standing in a circle. They are evenly spaced and consecutively numbered starting with 1. The student with number 3 is standing directly across form the student with number 17. How many students are there in Mrs. Pohl’s class?

  (3 pts) 4. $28$

Alice and Bill are walking in opposite directions along the same route between $A$ and $B$. Alice is going from $A$ to $B$, and Bill from $B$ to $A$. They start at the same time. They pass each other 3 hours later. Alice arrives at $B$ 2.5 hours before Bill arrives at $A$. How many hours does it take for Bill to go from $B$ to $A$?

  (3 pts) 5. $7.5$ hours

You have two boxes. Each of them has a square base and is half as tall as it is wide. If the larger box is two inches wider than the smaller box, and has a volume 244 in³ greater, what is the width of the smaller box?

  (3 pts) 6. $8$ in

In the triangle $\mathrm{\Delta }ABC$, the point $D$ lies on side $\overline{BC}$. Also, $AC=3$, $AD=3$, $BD=8$, and $CD=1$. What is AB?

  (3 pts) 7. $9$

If $x=2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{24}}+{\displaystyle \frac{1}{120}}+\mathrm{\dots }$, then $x$ is better known as

a) $\pi$      b) $e$      c) $\phi$     d) none of the previous choices

  (20 pts) 1.

Compute $\left[2^{2^2}\cdot 2\cdot {(2^2\cdot 2^2)}^2-{\left({\displaystyle \genfrac{}{}{0pt}{}{2^2+2^{2-2}}{2}}\right)}^2\right]/2^2$.

  (20 pts) 2.

Given a whole number construct a sequence as follows. If the number is even, divide by 2; if it’s odd multiply by three and add 1. Apply the same method to the result to get the next number. Stop if/when you reach 1. How many steps are required if you start with 112?

  (20 pts) 3.

In the decimal expansion of $1/14$ what is the value of the 30th digit to the right of the decimal point?

  (20 pts) 4.

Find a positive integer $m$ so that

  (20 pts) 5.

If $f_0=0,f_1=1$ and for $n\ge 2,f_n=f_{n-1}+f_{n-2}$ find $f_4+f_9+f_{16}$.

  (20 pts) 6.

Find a ten digit number which contains every digit 0,1,…, 9 exactly once, starts with a 3 and is divisible by every whole number between 2 and 18.

  (20 pts) 7.

Starting with $x_0=1$, for $n\ge 1$ let $x_n$ be the least non-negative remainder when $2\cdot x_{n-1}$ is divided by $n+6$. Find the smallest value of $n$ for which $x_n=0$.

  (20 pts) 8.

Factor $11687$ into primes.

  (20 pts) 9.

If $y=\cos \left({\displaystyle \frac{7\pi }{12}}\right)$, then the value of $y$ is

a) ${\displaystyle \frac{\sqrt{2}-\sqrt{6}}{4}}$ b) ${\displaystyle \frac{-\sqrt{2-\sqrt{3}}}{2}}$ c) approximately $-0.258819$

d) all of the previous choices

  (20 pts) 10.

If $x=2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{24}}+{\displaystyle \frac{1}{120}}+\mathrm{\dots }$, then $x$ is better known as

a) $\pi$      b) $e$      c) $\phi$     d) none of the previous choices

  (20 pts) 1. b.

Compute $\left[2^{2^2}\cdot 2\cdot {(2^2\cdot 2^2)}^2-{\left({\displaystyle \genfrac{}{}{0pt}{}{2^2+2^{2-2}}{2}}\right)}^2\right]/2^2$.

  (20 pts) 2. $2023$

Given a whole number construct a sequence as follows. If the number is even, divide by 2; if it’s odd multiply by three and add 1. Apply the same method to the result to get the next number. Stop if/when you reach 1. How many steps are required if you start with 112?

  (20 pts) 3. $20$

In the decimal expansion of $1/14$ what is the value of the 30th digit to the right of the decimal point?

  (20 pts) 4. $8$

Find a positive integer $m$ so that

  (20 pts) 5. $134$

If $f_0=0,f_1=1$ and for $n\ge 2,f_n=f_{n-1}+f_{n-2}$ find $f_4+f_9+f_{16}$.

  (20 pts) 6. $1024$

Find a ten digit number which contains every digit 0,1,…, 9 exactly once, starts with a 3 and is divisible by every whole number between 2 and 18.

  (20 pts) 7. $3,785,942,160$

Starting with $x_0=1$, for $n\ge 1$ let $x_n$ be the least non-negative remainder when $2\cdot x_{n-1}$ is divided by $n+6$. Find the smallest value of $n$ for which $x_n=0$.

  (20 pts) 8. $18$

Factor $11687$ into primes.

  (20 pts) 9. $13\cdot 29\cdot 31$

If $y=\cos \left({\displaystyle \frac{7\pi }{12}}\right)$, then the value of $y$ is

a) ${\displaystyle \frac{\sqrt{2}-\sqrt{6}}{4}}$ b) ${\displaystyle \frac{-\sqrt{2-\sqrt{3}}}{2}}$ c) approximately $-0.258819$

d) all of the previous choices

  (20 pts) 10. d.

If $x=2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{24}}+{\displaystyle \frac{1}{120}}+\mathrm{\dots }$, then $x$ is better known as

a) $\pi$      b) $e$      c) $\phi$     d) none of the previous choices

The answer is b.

Compute $\left[2^{2^2}\cdot 2\cdot {(2^2\cdot 2^2)}^2-{\left({\displaystyle \genfrac{}{}{0pt}{}{2^2+2^{2-2}}{2}}\right)}^2\right]/2^2$.

The answer is $2023$ by straightforward calculation.

Given a whole number construct a sequence as follows. If the number is even, divide by 2; if it’s odd multiply by three and add 1. Apply the same method to the result to get the next number. Stop if/when you reach 1. How many steps are required if you start with 112?

$112⟶56⟶28⟶14⟶7⟶22⟶11⟶34⟶17⟶52⟶26⟶13⟶40⟶20⟶10⟶5⟶16⟶8⟶4⟶2⟶1$. There are $20$ arrows.

In the decimal expansion of $1/14$ what is the value of the 30th digit to the right of the decimal point?

The expansion starts $0.0714285714285\mathrm{\dots }$. Since the 6th digit is 8 and the cycle length is 6, the answer is 8.

Find a positive integer $m$ so that $1+2+3+\mathrm{\cdots }+90=100+101+\mathrm{\cdots }+m.$

The answer is $m=134$ which can be seen by using Gauss’ formula for $1+2+3+\mathrm{\dots }+n$ three times after adding $1+2+\mathrm{\dots }+99$ to both sides.

If $f_0=0,f_1=1$ and for $n\ge 2,f_n=f_{n-1}+f_{n-2}$ find $f_4+f_9+f_{16}$.

Using the recursive formula one computes $f_4+f_9+f_{16}=3+34+987=1024$

Find a ten digit number which contains every digit 0,1,…, 9 exactly once, starts with a 3 and is divisible by every whole number between 2 and 18.

If y represents our number, y is divisible by $2^2\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17=12,252,240$. So $y=12,252,240x$, where $245\le x\le 326$ since $3,000,000,000\le y\le 4,000,000,000$. Trial and error leads to $x=309$ and $y=3,785,942,160$.

Starting with $x_0=1$, for $n\ge 1$ let $x_n$ be the least non-negative remainder when $2\cdot x_{n-1}$ is divided by $n+6$. Find the smallest value of $n$ for which $x_n=0$.

One finds the sequence to be $1,2,4,8,6,1,2,4,8,1,2,4,8,16,12,3,6,12,0$ so $n=18$.

Factor $11687$ into primes.

The answer is $11687=13\cdot 29\cdot 31$.

If $y=\cos \left({\displaystyle \frac{7\pi }{12}}\right)$, then $y$ is

a) ${\displaystyle \frac{\sqrt{2}-\sqrt{6}}{4}}$    b) ${\displaystyle \frac{-\sqrt{2-\sqrt{3}}}{2}}$    c) approximately $-0.258819$   d) all of the previous choices

The answer is d)

A cube with edge length 8 feet was molded into a sphere. Find the diameter of the sphere.

  (20 pts) 1.

Find the value of $k$ if the graph of a linear function $y=f(x)$ has slope $m=3$ and passes through the points $(4,7)$ and $(k,1)$.

  (20 pts) 2.

The sum of the squares of two consecutive positive odd integers is 74. What is the value of the larger integer?

  (20 pts) 3.

Find the length of the arc of a circle of radius $10$ inches subtended by a central angle of ${50}^{\circ }$.

  (20 pts) 4.

Suppose the rational function $f(x)$ is given by $f(x)={\displaystyle \frac{5x+7}{2x+4}}$. Find $f^{-1}(x)$.

  (20 pts) 5.

The London Eye is a huge Ferris wheel with a diameter of $135$ meters ($443$ feet). It completes one rotation every $30$ minutes. Riders board from a platform $2$ meters above the ground. Express a rider’s height above ground as a function of time in minutes after boarding.

  (20 pts) 6.

If the cost of a bat and a baseball combined is $1.10 and the ball costs $1.00 less than the bat, how much is the ball?

  (20 pts) 7.

Let ${\mathrm{\ell }}_1$ be the line segment whose endpoints lie at the top left and bottom right corner of a square and let ${\mathrm{\ell }}_2$ be the line segment whose endpoints are the bottom left corner of the square and the center of the top side of the square. What fraction of the area of the square is not occupied by the triangle formed from the point at the intersection of ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$ and the bottom two corners of the square?

  (20 pts) 8.

The square root of the value obtained by adding twelve to some positive number is the same as the number itself. What is it?

  (20 pts) 9.

Suppose a town has a population of $1,000$ in the year $1999$, and suppose the population of the town grows at a constant rate of $125$ per year. Find the equation of the line that models the town’s population $P$ as a function of $t$, where $t$ is the number of years since the model began.

  (20 pts) 10.

A cube with edge length 8 feet was molded into a sphere. Find the diameter of the sphere.

  (20 pts) 1. $8\sqrt[3]{\frac{6}{\pi }}$

Find the value of $k$ if the graph of a linear function $y=f(x)$ has slope $m=3$ and passes through the points $(4,7)$ and $(k,1)$.

  (20 pts) 2. $k=2$

The sum of the squares of two consecutive positive odd integers is 74. What is the value of the larger integer?

  (20 pts) 3. $7$

Find the length of the arc of a circle of radius $10$ inches subtended by a central angle of ${50}^{\circ }$.

  (20 pts) 4. ${\displaystyle \frac{25\pi }{9}}$inches

Suppose the rational function $f(x)$ is given by $f(x)={\displaystyle \frac{5x+7}{2x+4}}$. Find $f^{-1}(x)$.

  (20 pts) 5. $f^{-1}(x)={\displaystyle \frac{7-4x}{2x-5}}$

The London Eye is a huge Ferris wheel with a diameter of $135$ meters ($443$ feet). It completes one rotation every $30$ minutes. Riders board from a platform $2$ meters above the ground. Express a rider’s height above ground as a function of time in minutes after boarding.

  (20 pts) 6. $h(t)=-67.5\cos \left(\frac{\pi t}{15}\right)+69.5$ or $h(t)=-67.5\sin \left(\frac{\pi }{2}-\frac{\pi }{15}t\right)+69.5$ or $h(t)=67.5\sin \left(\frac{\pi }{15}t-\frac{\pi }{2}\right)+69.5$

If the cost of a bat and a baseball combined is $1.10 and the ball costs $1.00 less than the bat, how much is the ball?

  (20 pts) 7. $0.05

Let ${\mathrm{\ell }}_1$ be the line segment whose endpoints lie at the top left and bottom right corner of a square and let ${\mathrm{\ell }}_2$ be the line segment whose endpoints are the bottom left corner of the square and the center of the top side of the square. What fraction of the area of the square is not occupied by the triangle formed from the point at the intersection of ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_2$ and the bottom two corners of the square?

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

The square root of the value obtained by adding twelve to some positive number is the same as the number itself. What is it?

  (20 pts) 9. $4$

Suppose a town has a population of $1,000$ in the year $1999$, and suppose the population of the town grows at a constant rate of $125$ per year. Find the equation of the line that models the town’s population $P$ as a function of $t$, where $t$ is the number of years since the model began.

  (20 pts) 10. $P=125t+1,000$