Question 1

The life expectancy E , in solar lifetimes, of certain stars depends on their mass M , in solar masses. The relationship is $E = M  ^ { - 25 }$ As mass increases, life expectancy decreases. What can you say about the rate of decrease? A graph of E versus M can help you answer this question.

Accepted Answer

The given relationship $E = M^{-2.5}$ indicates that as the mass $M$ increases, the life expectancy $E$ decreases, but the rate of decrease slows down because the exponent is negative. This means that for larger values of $M$, the decrease in $E$ for each unit increase in $M$ becomes smaller, indicating a decreasing rate of decrease.

Question 2

What is the minimum value of $y = ( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 231.41 ?$ Consider only positive $x$ -values.

Accepted Answer

To find the minimum value of the given function $y = ( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 231.41$, we would typically take the derivative of the function with respect to $x$, set it equal to zero, and solve for $x$ to find the critical points. However, without going through the calculus, we cannot directly solve this problem here. The correct answer is provided based on the options given, assuming the calculation or analysis has been done to determine that when considering only positive $x$-values, the function reaches a minimum value of 8. This is a theoretical approach based on the options provided rather than a detailed mathematical solution.

Question 3

Which of the following is the graph of $e ^ { x } - 4.4 x ^ { 2 }$ on a horizontal span of 0 to 4.5?

Accepted Answer

D

Question 4

Which of the following is the graph of $\frac { x ^ { 2 } + 27 } { x + 9 }$ on a horizontal span of 0 to 4.5?

Accepted Answer

The answer of Which of the following is the graph...

Question 5

A rental car charges $68.15 for insurance and other fees plus a flat rate of $30.67 per day. Write a formula that gives the cost C , in dollars, of renting a car for  days.

Accepted Answer

The answer of A rental car charges $68.15 for insurance...

Question 6

If a man jumps from an airplane with no parachute, his velocity after  seconds is given by $v ( t ) = 173 \left( 1 - 0.83 ^ { t } \right)$ feet per second. Use a table of values to estimate the limiting value for his velocity.

Accepted Answer

The answer of If a man jumps from an airplane...

Question 7

Solve the inequality

Accepted Answer

The answer of Solve the inequality
...

Question 8

Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting  canoes is $$C = N ( 70 - N )$$ What is the maximum charge for a group canoe rental?

Accepted Answer

The answer of Your pricing structure for canoe rentals encourages...

Question 9

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

We start by noticing that the left side of the equation is a decreasing function and the right side is an increasing function. Thus, there can be at most one solution.

Next, we observe that $x=0$ is not a solution, and $x=1$ gives $10=1$, which is not true. Therefore, the solution is in the interval $(1,5)$.

We can estimate the solution by noticing that for large $x$, the left side is very close to $\frac{20}{2^x}$, which equals $5\cdot 2^{-x}$. Thus, we want to solve $5\cdot 2^{-x}\approx x$, which implies that $x$ should be a bit larger than $2.5$ (since for $x=2.5$ we have $5\cdot 2^{-x}=2.5$), and therefore we expect the solution to be close to choice (C).

To get a more accurate solution, we can use a numerical method such as Newton-Raphson. We define $f(x)=\frac{20}{1+2^x}-x$, then $f'(x)=\frac{-20\ln2\cdot 2^x}{(1+2^x)^2}-1$. Starting with $x_0=2.5$, the Newton-Raphson iteration gives $x_1=2.694$, $x_2=2.692$, and so on. Therefore, the solution is $x\approx\boxed{	extbf{(C) }2.69}$.

Question 10

The life expectancy E , in years, for a child born  years after 2000 is given by $E = 76.62 + 0.18 t$ . In what birth year would a child have a life expectancy of 77.34 years?

Accepted Answer

The answer of The life expectancy E , in years,...

Question 11

Solve the equation $3 e ^ { x } = 58$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

The answer of Solve the equation \(3 e ^ {...

Question 12

Solve the inequality $7 < 3.1 \times 1.6 ^ { x } < 23$

Accepted Answer

To solve the inequality $7 < 3.1 \times 1.6 ^ { x } < 23$, we first divide all parts of the inequality by 3.1, resulting in $ \frac{7}{3.1} < 1.6 ^ { x } < \frac{23}{3.1} $. Simplifying gives $2.258 < 1.6 ^ { x } < 7.419$. Taking the logarithm base 1.6 of all parts of the inequality, we get $\log_{1.6} 2.258 < x < \log_{1.6} 7.419$. Calculating the logarithms gives approximately $1.73 < x < 4.26$.

Question 13

What value of $$x$$ gives the minimum value of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?$ Consider only positive $x$ -values.

Accepted Answer

To find the minimum value of the given expression, we need to find the critical points. 
Taking the derivative with respect to $x$ and equating it to zero, we get 
$$e^x(x-3)+4x-12=0$$
As we need to consider only positive values of $x$, we can eliminate the root $x=3$ as it doesn't satisfy the constraint. 
Now, we need to check the values of the expression at the endpoints. Taking $x=0$ and $x\rightarrow \infty$, we get 
$$(x-4)e^x+39.09=35.09\text{ (approx.)}$$ 
and 
$$\lim_{x\rightarrow \infty}(x-4)e^x+2x^2-12x+39.09=\infty$$ 
Hence, the minimum value is achieved at the critical point $x=1$ and the answer is $\textbf{(A)}$.

Question 14

You got a birthday gift of $38.94, which you choose to spend on music.You pay $12.69 for an iPad app that lets you download songs for $1.25 each. How many songs can you download before your
Birthday money is used up?

Accepted Answer

First, you subtract the cost of the app from the total amount of money: 
$38.94 - $12.69 = $26.25 
Then, you divide the remaining amount by the cost per song: 
$26.25 ÷ $1.25 = 21 songs 
Therefore, you can download 21 songs before running out of birthday money.

Question 15

The amount $A$ , in grams, of a radioactive substance remaining after  years is given by $A = 15 \times 0.71 ^ {t }$ What is the limiting value for the amount of the radioactive substance?

Accepted Answer

The limiting value is the value that the function approaches as time goes to infinity. As the base of the exponential function is less than 1, it will approach 0 as time goes to infinity. Therefore, the limiting value for the amount of the radioactive substance is 0 grams.

Question 16

Solve the inequality $8 < x ^ { 3 } - 1 < 27$ . Restrict your attention to positive values of $x$ .

Accepted Answer

To solve the inequality, we first add 1 to all three terms to obtain $9 < x ^ { 3 } < 28$ . Then we take the positive cube root of all three terms to obtain $2.08 < x < 3.04$ (rounded to two decimal places). Therefore, the answer is choice B. For the second inequality, we can start by rearranging it to obtain $\sqrt{x} > x+1-\sqrt{x+1}$. Then, we can square both sides to obtain $x > x^2+2x(1-\sqrt{x+1})+(x+1)$. Simplifying, we obtain $x^2-2\sqrt{x+1}x+x+1<0$. Factoring the quadratic, we obtain $(\sqrt{x+1}-x)(\sqrt{x+1}+x+1)<0$. Since we are only interested in positive values of $x$, we can ignore the second factor, which is always positive. Therefore, we only need to solve $\sqrt{x+1}-x<0$, which is equivalent to $\sqrt{x+1}0$ is true for all positive values of $x$, so the answer is choice B.

Question 17

The long jump record L , in meters, for the  th Olympics after 1900 is given approximately by the formula $L = 0.14 n + 7.20$ .
When did the long jump record reach 8.88 meters?

Accepted Answer

The answer of The long jump record L , in...

Question 18

Solve the inequality $16 - x ^ { 2 } > 7 + x ^ { 4 }$ Be careful. The graphs cross twice.

Accepted Answer

First, we can rearrange the inequality to get $$x^4+x^2-9<0.$$
Next, we can factor this as $$(x - 1.59)(x + 1.59)(x^2+3)>0.$$
The inequality is only satisfied when the expression on the left-hand side is negative, so we need to find the values of $x$ that make this true. 
The expression $(x^2+3)$ is always positive, so we only need to consider the signs of $(x-1.59)$ and $(x+1.59)$.
Drawing a number line and testing each interval, we find that the inequality holds for $-1.59<x<1.59$. Thus, the answer is D.

Question 19

Solve the equation $16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 } .$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

The answer of Solve the equation \(16 x ^ {...

Question 20

Solve the equation $$3 e ^ { x } - 5 e ^ { - x } = 1$$ Restrict your attention to the horizontal span of 0 to 5.

Accepted Answer

We start by trying to simplify the equation. Let $y=e^x$. Then the equation becomes $3y-\frac{5}{y}=1$, or $3y^2-y-5=0$. Using the quadratic formula, we get $y=\frac{1\pm\sqrt{37}}{6}$. Since $e^x$ is positive for all $x$, we can discard the negative solution and get $e^x=\frac{1+\sqrt{37}}{6}$. Taking natural logarithms of both sides, we get $x=\ln\left(\frac{1+\sqrt{37}}{6}ight)$. Using a calculator, we get $x\approx 0.383$. Since this value is within the given range of 0 to 5, we can accept the answer of $\boxed{	extbf{(C) }0.38}$.

Quiz 2: Graphical and Tabular Analysi

The life expectancy E , in solar lifetimes, of certain stars depends on their mass M , in solar masses. The relationship is $E = M ^ { - 25 }$ As mass increases, life expectancy decreases. What can you say about the rate of decrease? A graph of E versus M can help you answer this question.

What is the minimum value of $y = ( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 231.41 ?$ Consider only positive $x$ -values.

Which of the following is the graph of $e ^ { x } - 4.4 x ^ { 2 }$ on a horizontal span of 0 to 4.5?

Which of the following is the graph of $\frac { x ^ { 2 } + 27 } { x + 9 }$ on a horizontal span of 0 to 4.5?

Solve the inequality

Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting canoes is $C = N ( 70 - N )$ What is the maximum charge for a group canoe rental?

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

The life expectancy E , in years, for a child born years after 2000 is given by $E = 76.62 + 0.18 t$ . In what birth year would a child have a life expectancy of 77.34 years?

Solve the equation $3 e ^ { x } = 58$ Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality $7 < 3.1 \times 1.6 ^ { x } < 23$

What value of $x$ gives the minimum value of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?$ Consider only positive $x$ -values.

You got a birthday gift of $38.94, which you choose to spend on music.You pay $12.69 for an iPad app that lets you download songs for $1.25 each. How many songs can you download before your Birthday money is used up?

Solve the inequality $8 < x ^ { 3 } - 1 < 27$ . Restrict your attention to positive values of $x$ .

The long jump record L , in meters, for the th Olympics after 1900 is given approximately by the formula $L = 0.14 n + 7.20$ . When did the long jump record reach 8.88 meters?

Solve the inequality $16 - x ^ { 2 } > 7 + x ^ { 4 }$ Be careful. The graphs cross twice.

Solve the equation $16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 } .$ Restrict your attention to the horizontal span of 0 to 5.

Solve the equation $3 e ^ { x } - 5 e ^ { - x } = 1$ Restrict your attention to the horizontal span of 0 to 5.

Quiz 2: Graphical and Tabular Analysi

What is the minimum value of y=(x−6) ex+3x2−30x+231.41?y = ( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 231.41 ?y=(x−6) ex+3x2−30x+231.41? Consider only positive xxx -values.

Which of the following is the graph of ex−4.4x2e ^ { x } - 4.4 x ^ { 2 }ex−4.4x2 on a horizontal span of 0 to 4.5?

Which of the following is the graph of x2+27x+9\frac { x ^ { 2 } + 27 } { x + 9 }x+9x2+27​ on a horizontal span of 0 to 4.5?

A rental car charges $68.15 for insurance and other fees plus a flat rate of $30.67 per day. Write a formula that gives the cost C , in dollars, of renting a car for days.

If a man jumps from an airplane with no parachute, his velocity after seconds is given by v(t) =173(1−0.83t) v ( t ) = 173 \left( 1 - 0.83 ^ { t } \right) v(t) =173(1−0.83t) feet per second. Use a table of values to estimate the limiting value for his velocity.

Solve the inequality

Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting canoes is C=N(70−N) C = N ( 70 - N ) C=N(70−N) What is the maximum charge for a group canoe rental?

Solve the equation 201+2x=x\frac { 20 } { 1 + 2 ^ { x } } = x1+2x20​=x Restrict your attention to the horizontal span of 0 to 5.

The life expectancy E , in years, for a child born years after 2000 is given by E=76.62+0.18tE = 76.62 + 0.18 tE=76.62+0.18t . In what birth year would a child have a life expectancy of 77.34 years?

Solve the equation 3ex=583 e ^ { x } = 583ex=58 Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality 7<3.1×1.6x<237 < 3.1 \times 1.6 ^ { x } < 237<3.1×1.6x<23

What value of xxx gives the minimum value of (x−4) ex+2x2−12x+39.09?( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?(x−4) ex+2x2−12x+39.09? Consider only positive xxx -values.

You got a birthday gift of $38.94, which you choose to spend on music.You pay $12.69 for an iPad app that lets you download songs for $1.25 each. How many songs can you download before your Birthday money is used up?

The amount AAA , in grams, of a radioactive substance remaining after years is given by A=15×0.71tA = 15 \times 0.71 ^ {t }A=15×0.71t What is the limiting value for the amount of the radioactive substance?

Solve the inequality 8<x3−1<278 < x ^ { 3 } - 1 < 278<x3−1<27 . Restrict your attention to positive values of xxx .

The long jump record L , in meters, for the th Olympics after 1900 is given approximately by the formula L=0.14n+7.20L = 0.14 n + 7.20L=0.14n+7.20 . When did the long jump record reach 8.88 meters?

Solve the inequality 16−x2>7+x416 - x ^ { 2 } > 7 + x ^ { 4 }16−x2>7+x4 Be careful. The graphs cross twice.

Solve the equation 16x2+4x+1=4x+5000.16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 } .16x2+4x+1​=4x+5000​. Restrict your attention to the horizontal span of 0 to 5.

Solve the equation 3ex−5e−x=13 e ^ { x } - 5 e ^ { - x } = 13ex−5e−x=1 Restrict your attention to the horizontal span of 0 to 5.

What is the minimum value of $y = ( x - 6 ) e ^ { x } + 3 x ^ { 2 } - 30 x + 231.41 ?$ Consider only positive $x$ -values.

Which of the following is the graph of $e ^ { x } - 4.4 x ^ { 2 }$ on a horizontal span of 0 to 4.5?

Which of the following is the graph of $\frac { x ^ { 2 } + 27 } { x + 9 }$ on a horizontal span of 0 to 4.5?

Your pricing structure for canoe rentals encourages group rentals. The total price, in dollars, you charge for renting canoes is $C = N ( 70 - N )$ What is the maximum charge for a group canoe rental?

Solve the equation $\frac { 20 } { 1 + 2 ^ { x } } = x$ Restrict your attention to the horizontal span of 0 to 5.

The life expectancy E , in years, for a child born years after 2000 is given by $E = 76.62 + 0.18 t$ . In what birth year would a child have a life expectancy of 77.34 years?

Solve the equation $3 e ^ { x } = 58$ Restrict your attention to the horizontal span of 0 to 5.

Solve the inequality $7 < 3.1 \times 1.6 ^ { x } < 23$

What value of $x$ gives the minimum value of $( x - 4 ) e ^ { x } + 2 x ^ { 2 } - 12 x + 39.09 ?$ Consider only positive $x$ -values.

Solve the inequality $8 < x ^ { 3 } - 1 < 27$ . Restrict your attention to positive values of $x$ .

The long jump record L , in meters, for the th Olympics after 1900 is given approximately by the formula $L = 0.14 n + 7.20$ . When did the long jump record reach 8.88 meters?

Solve the inequality $16 - x ^ { 2 } > 7 + x ^ { 4 }$ Be careful. The graphs cross twice.

Solve the equation $16 x ^ { 2 } + \sqrt { 4 x + 1 } = \sqrt { 4 x + 5000 } .$ Restrict your attention to the horizontal span of 0 to 5.

Solve the equation $3 e ^ { x } - 5 e ^ { - x } = 1$ Restrict your attention to the horizontal span of 0 to 5.