Question 1

Aretailer that n games can be sold in a month at a price of 20-0.1n dollars per game. Assume that he buys each game for $13, and sells every one that he buys. if he whishes to make a profit of at least $100 per month on sales of this game, how many games must he sell each month?

Accepted Answer

To find the number of games (n) the retailer must sell to make at least $100 profit, we first determine the profit per game, then find the total profit, and solve for n. The selling price per game is $20-0.1n, and the cost is $13, so the profit per game is $(20-0.1n) - $13 = $7-0.1n. The total profit for n games is n($7-0.1n). Setting this equal to $100 (the minimum desired profit) gives us the equation n($7-0.1n) = $100. Simplifying, we get $7n - 0.1n^2 = $100. Rearranging, we get $0.1n^2 - 7n + 100 = 0. Solving this quadratic equation for n gives us the range of n that satisfies the profit requirement. The correct range that meets the condition for a profit of at least $100 is 20 ≤ n ≤ 50.

Question 2

$\text { Solve } f ( x ) \leq g ( x )$

Accepted Answer

The answer of \(\text { Solve } f ( x...

Question 3

Assume that the elevation  E , in feet, of a sag in a proposed route is given by $E ( x ) = 0.000039 x ^ { 2 } - 0.25 x + 1300$ where  x  represents the horizontal distance in feet along the proposed route and$$0 \leq x \leq 5000$$ . For what  x-values  is the elevation 1100 feet or more? Round your answer to the nearest foot.

Accepted Answer

The answer of Assume that the elevation  E ,...

Question 4

The cost  C  of producing  t  units is given by $C ( t ) = 3 t ^ { 2 } + 7 t$ and the revenue  R  generated from selling  t  units is given by $R ( t ) = 4 t ^ { 2 } + t$. For what values of  t  will there be a profit?

Accepted Answer

Profit occurs when revenue exceeds cost, so we set $R(t) > C(t)$ and solve for $t$. This gives $4t^2 + t > 3t^2 + 7t$, simplifying to $t^2 - 6t > 0$, which factors to $t(t - 6) > 0$. This inequality is satisfied for $t > 6$.

Question 5

Assume that the profit P made when  t  units are sold, $t > 0$, is given by $P ( t ) = t ^ { 2 } - 18 t + 72$ For what values of  t  will there be a profit (that is, $\mathrm { P } > 0 )$?

Accepted Answer

The profit is positive when $P(t) > 0$. Factoring, we get $P(t) = (t - 6)(t - 12)$. So, the profit is positive when $6 < t < 12$ or $t > 12$.

Question 6

The Fahrenheit temperature,  F , in Siber City in October ranges from  $71 ^ { \circ } \mathrm { F } \text { to } 43 ^ { \circ } \mathrm { F }$.  Write an absolute value inequality whose solution is this range.

Accepted Answer

The correct absolute value inequality is $| \mathrm { F } - 57 | < 14$. This is because the average of the maximum and minimum temperatures (71 and 43) is 57, and the difference between the average and either the maximum or minimum temperature is 14. This inequality represents all temperatures within 14 degrees of 57, which matches the given range.

Question 7

Assume that the profit P made when  t  units are sold, $t > 0$ is given by $P ( t ) = t ^ { 2 } - 28 t + 180$ For what values of t will there be a loss (that is, $\mathrm { P } < 0 \}$ ?

Accepted Answer

To determine when there is a loss, we need to find when $P(t) < 0$. Factoring the quadratic equation $P(t) = t^2 - 28t + 180$, we get $(t-10)(t-18) < 0$. This inequality is true for $t$ values between 10 and 18, hence $10 < t < 18$.

Question 8

Assume that from 1930 to 1980 , the number of deaths from lead poisoning (per 100,000 people) decreased dramatically and could be modeled by $\mathrm { f } ( \mathrm { x } ) = - 0.0148 \mathrm { x } ^ { 2 } + 54.2 \mathrm { x } - 49,200$, where $\mathrm { x }$ is the year. Determine when, to the nearest year, this death rate was 150 or less by graphing $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ and $\mathrm { y } = 150$ in $[ 1930,1980,10 ]$ by $[ 0,200$, 25 ].

Accepted Answer

The function $ f(x) $ represents a quadratic equation, and by graphing it along with $ y = 150 $, we find that the death rate was 150 or less starting in 1967.

Question 9

A rectangular enclosure must have an area of at least 2400yd² . If 220yd of fencing is to be used, and the width cannot exceed the length, within what limits must the width of the enclosure lie?

Accepted Answer

The perimeter of the rectangle is 2(length + width) = 220 yards, so length + width = 110 yards. Since the area is length * width ≥ 2400 square yards and width cannot exceed length, the width must be at least 30 yards to ensure the area is at least 2400 square yards (since 30*80 = 2400 and 30+80=110). The maximum width is when width equals length, so width = 110/2 = 55 yards. Therefore, the width must lie between 30 and 55 yards.

Question 10

The height  h  of a ball thrown is given by $h ( x ) = 2 x - 0.08 x ^ { 2 }$ where  x  is the horizontal distance traveled and both measurements are in feet. At what horizontal distances will the height of the ball be more than  9 ft  ? (Round distances to the nearest tenth of a foot, if necessary.)

Accepted Answer

The quadratic equation $h(x) = 2x - 0.08x^2$ is set to be greater than 9 to find the range of x. Solving $2x - 0.08x^2 > 9$ gives the interval $5.9 \mathrm{ft} < x < 19.1 \mathrm{ft}$.

Question 11

The graphs of two functions f(x) and g(x) are given.
-Solve $f ( x ) \geq g ( x )$.

Accepted Answer

The answer of The graphs of two functions f(x) and...

Question 12

A flare fired from the bottom of a gorge is visible only when the flare is above the rim. If it is fired with an initial velocity of  112 ft/sec  and the gorge is  160 ft deep, during what interval can the flare be seen?
$\left( h ( t ) = - 16 t ^ { 2 } + v 0 t + h 0 . \right)$

Accepted Answer

The answer of A flare fired from the bottom of...

Question 13

The height h in feet of a frisbee thrown is given by $$h ( x ) = x - 0.099 x ^ { 2 } + 3$$ , where x is the time traveled in seconds. The frisbee is released at a height of 3 feet. At what time will the height of the frisbee be 3 feet again?
(Round your answer to the nearest tenth of a second, if necessary.)

Accepted Answer

The answer of The height h in feet of a...

Question 14

The graphs of two functions f(x) and g(x) are given.
-Solve $f ( x ) \leq g ( x )$.

Accepted Answer

The answer of The graphs of two functions f(x) and...

Question 15

The graphs of two functions f(x) and g(x) are given.
-Solve $f ( x ) \leq g ( x )$.

Accepted Answer

The answer of The graphs of two functions f(x) and...

Question 16

A coin is tossed upward from a balcony 390 ft high (h₀) with an initial velocity (v₀) of 16 ft/sec , according to the formula $h ( t ) = - 16 t ^ { 2 } + v _ { 0 } t + h 0$. where t is time in seconds. During what interval of time will the coin be at a height of at least 70 ft ?

Accepted Answer

To find the interval of time when the coin is at least 70 ft high, we substitute $h(t) = 70$ into the given formula and solve for $t$. The equation becomes $70 = -16t^2 + 16t + 390$. Rearranging gives $16t^2 - 16t - 320 = 0$, which simplifies to $t^2 - t - 20 = 0$. Factoring or using the quadratic formula yields two time values, which represent the times when the coin is exactly 70 ft high during its ascent and descent. The interval between these times, including the initial toss and the moment it reaches 70 ft on its way down, covers from $0 \sec$ to approximately $5 \sec$, indicating the coin is at or above 70 ft within this time frame.

Question 17

If a rocket is propelled upward from ground level, its height  h  in meters after  t  sec is given by  h(t)=$- 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }$ During what interval of time will the rocket be higher than  205.8 m  ?

Accepted Answer

To find when the rocket is higher than 205.8 m, we set $h(t) > 205.8$ and solve for $t$. Substituting $h(t) = -9.8t^2 + 98t$, we get $-9.8t^2 + 98t > 205.8$. Rearranging, we have $-9.8t^2 + 98t - 205.8 > 0$. Solving this quadratic inequality for $t$, we find the time interval where the rocket's height exceeds 205.8 m is between 3 and 7 seconds.

Question 18

Solve $f ( x ) \geq g ( x$

Accepted Answer

The answer of Solve \(f ( x ) \geq g...

Question 19

The differance between thr tuition at State University and Highbrow Collage is less than $900. The tuition at Highbrow is $5000. Express this as an absolute value inequality. Use "x" for the tuition of state University.

Accepted Answer

The absolute value inequality $$| x - 5000 | < 900$$ correctly represents the situation where the difference between the tuition at State University (x) and Highbrow College ($5000) is less than $900, regardless of which is higher.

Quiz 4: Additional Topics With Functions

Aretailer that n games can be sold in a month at a price of 20-0.1n dollars per game. Assume that he buys each game for $13, and sells every one that he buys. if he whishes to make a profit of at least $100 per month on sales of this game, how many games must he sell each month?

$\text { Solve } f ( x ) \leq g ( x )$ $\text { Solve } f ( x ) \leq g ( x ) A) x \leq - 4 \text { or } x \geq 4 B) x \leq - 1 \text { or } x \geq 1 C) - 1 \leq x \leq 1 D) - 4 \leq x \leq 4$

The cost C of producing t units is given by $C ( t ) = 3 t ^ { 2 } + 7 t$ and the revenue R generated from selling t units is given by $R ( t ) = 4 t ^ { 2 } + t$ . For what values of t will there be a profit?

Assume that the profit P made when t units are sold, $t > 0$ , is given by $P ( t ) = t ^ { 2 } - 18 t + 72$ For what values of t will there be a profit (that is, $\mathrm { P } > 0 )$ ?

The Fahrenheit temperature, F , in Siber City in October ranges from $71 ^ { \circ } \mathrm { F } \text { to } 43 ^ { \circ } \mathrm { F }$ . Write an absolute value inequality whose solution is this range.

Assume that the profit P made when t units are sold, $t > 0$ is given by $P ( t ) = t ^ { 2 } - 28 t + 180$ For what values of t will there be a loss (that is, $\mathrm { P } < 0 \}$ ?

A rectangular enclosure must have an area of at least 2400yd² . If 220yd of fencing is to be used, and the width cannot exceed the length, within what limits must the width of the enclosure lie?

The height h of a ball thrown is given by $h ( x ) = 2 x - 0.08 x ^ { 2 }$ where x is the horizontal distance traveled and both measurements are in feet. At what horizontal distances will the height of the ball be more than 9 ft ? (Round distances to the nearest tenth of a foot, if necessary.)

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \geq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \geq g ( x ) . A) - 3 \leq x \leq 3 B) x \leq - 3 or x \geq 3 C) x \leq - 2 or x \geq 2 D) - 2 \leq x \leq 2$

A flare fired from the bottom of a gorge is visible only when the flare is above the rim. If it is fired with an initial velocity of 112 ft/sec and the gorge is 160 ft deep, during what interval can the flare be seen? $\left( h ( t ) = - 16 t ^ { 2 } + v 0 t + h 0 . \right)$

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \leq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \leq g ( x ) . A) - 3 \leq x \leq 3 B) - 2 \leq x \leq 2 C) x \leq - 3 or x \geq 3 D) x \leq - 2 or x \geq 2$

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \leq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \leq g ( x ) . A) x \leq - 1 or 1 \leq x \leq 4 B) - 1 \leq x \leq 4 C) x \leq 2 or 1 \leq x \leq 4 D) 1 \leq x \leq 4$

A coin is tossed upward from a balcony 390 ft high (h₀) with an initial velocity (v₀) of 16 ft/sec , according to the formula $h ( t ) = - 16 t ^ { 2 } + v _ { 0 } t + h 0$ . where t is time in seconds. During what interval of time will the coin be at a height of at least 70 ft ?

If a rocket is propelled upward from ground level, its height h in meters after t sec is given by h(t) = $- 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }$ During what interval of time will the rocket be higher than 205.8 m ?

Solve $f ( x ) \geq g ( x$ $Solve f ( x ) \geq g ( x A) - 1 \leq x \leq 1 B) x \leq - 6 \text { or } x \geq 6 C) x \leq - 1 \text { or } x \geq 1 D) - 6 \leq x \leq 6$

The differance between thr tuition at State University and Highbrow Collage is less than $900. The tuition at Highbrow is $5000. Express this as an absolute value inequality. Use "x" for the tuition of state University.

Quiz 4: Additional Topics With Functions

Aretailer that n games can be sold in a month at a price of 20-0.1n dollars per game. Assume that he buys each game for $13, and sells every one that he buys. if he whishes to make a profit of at least $100 per month on sales of this game, how many games must he sell each month?

Solve f(x) ≤g(x) \text { Solve } f ( x ) \leq g ( x ) Solve f(x) ≤g(x)

The cost C of producing t units is given by C(t) =3t2+7tC ( t ) = 3 t ^ { 2 } + 7 tC(t) =3t2+7t and the revenue R generated from selling t units is given by R(t) =4t2+tR ( t ) = 4 t ^ { 2 } + tR(t) =4t2+t . For what values of t will there be a profit?

Assume that the profit P made when t units are sold, t>0t > 0t>0 , is given by P(t) =t2−18t+72P ( t ) = t ^ { 2 } - 18 t + 72P(t) =t2−18t+72 For what values of t will there be a profit (that is, P>0) \mathrm { P } > 0 ) P>0) ?

The Fahrenheit temperature, F , in Siber City in October ranges from 71∘F to 43∘F71 ^ { \circ } \mathrm { F } \text { to } 43 ^ { \circ } \mathrm { F }71∘F to 43∘F . Write an absolute value inequality whose solution is this range.

Assume that the profit P made when t units are sold, t>0t > 0t>0 is given by P(t) =t2−28t+180P ( t ) = t ^ { 2 } - 28 t + 180P(t) =t2−28t+180 For what values of t will there be a loss (that is, P<0}\mathrm { P } < 0 \}P<0} ?

A rectangular enclosure must have an area of at least 2400yd2 . If 220yd of fencing is to be used, and the width cannot exceed the length, within what limits must the width of the enclosure lie?

The graphs of two functions f(x) and g(x) are given. -Solve f(x) ≥g(x) f ( x ) \geq g ( x ) f(x) ≥g(x) .

The graphs of two functions f(x) and g(x) are given. -Solve f(x) ≤g(x) f ( x ) \leq g ( x ) f(x) ≤g(x) .

The graphs of two functions f(x) and g(x) are given. -Solve f(x) ≤g(x) f ( x ) \leq g ( x ) f(x) ≤g(x) .

If a rocket is propelled upward from ground level, its height h in meters after t sec is given by h(t) = −9.8t2+98t- 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }−9.8t2+98t During what interval of time will the rocket be higher than 205.8 m ?

Solve f(x) ≥g(xf ( x ) \geq g ( xf(x) ≥g(x

The differance between thr tuition at State University and Highbrow Collage is less than $900. The tuition at Highbrow is $5000. Express this as an absolute value inequality. Use "x" for the tuition of state University.

$\text { Solve } f ( x ) \leq g ( x )$ $\text { Solve } f ( x ) \leq g ( x ) A) x \leq - 4 \text { or } x \geq 4 B) x \leq - 1 \text { or } x \geq 1 C) - 1 \leq x \leq 1 D) - 4 \leq x \leq 4$

The cost C of producing t units is given by $C ( t ) = 3 t ^ { 2 } + 7 t$ and the revenue R generated from selling t units is given by $R ( t ) = 4 t ^ { 2 } + t$ . For what values of t will there be a profit?

Assume that the profit P made when t units are sold, $t > 0$ , is given by $P ( t ) = t ^ { 2 } - 18 t + 72$ For what values of t will there be a profit (that is, $\mathrm { P } > 0 )$ ?

The Fahrenheit temperature, F , in Siber City in October ranges from $71 ^ { \circ } \mathrm { F } \text { to } 43 ^ { \circ } \mathrm { F }$ . Write an absolute value inequality whose solution is this range.

Assume that the profit P made when t units are sold, $t > 0$ is given by $P ( t ) = t ^ { 2 } - 28 t + 180$ For what values of t will there be a loss (that is, $\mathrm { P } < 0 \}$ ?

A rectangular enclosure must have an area of at least 2400yd² . If 220yd of fencing is to be used, and the width cannot exceed the length, within what limits must the width of the enclosure lie?

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \geq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \geq g ( x ) . A) - 3 \leq x \leq 3 B) x \leq - 3 or x \geq 3 C) x \leq - 2 or x \geq 2 D) - 2 \leq x \leq 2$

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \leq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \leq g ( x ) . A) - 3 \leq x \leq 3 B) - 2 \leq x \leq 2 C) x \leq - 3 or x \geq 3 D) x \leq - 2 or x \geq 2$

The graphs of two functions f(x) and g(x) are given. -Solve $f ( x ) \leq g ( x )$ . $The graphs of two functions f(x) and g(x) are given. -Solve f ( x ) \leq g ( x ) . A) x \leq - 1 or 1 \leq x \leq 4 B) - 1 \leq x \leq 4 C) x \leq 2 or 1 \leq x \leq 4 D) 1 \leq x \leq 4$

If a rocket is propelled upward from ground level, its height h in meters after t sec is given by h(t) = $- 9.8 \mathrm { t } ^ { 2 } + 98 \mathrm { t }$ During what interval of time will the rocket be higher than 205.8 m ?

Solve $f ( x ) \geq g ( x$ $Solve f ( x ) \geq g ( x A) - 1 \leq x \leq 1 B) x \leq - 6 \text { or } x \geq 6 C) x \leq - 1 \text { or } x \geq 1 D) - 6 \leq x \leq 6$