Question 1

Daily production levels in a plant can be modeled by the function $G(t)=-3 t^{2}$$+18t-9$ , which gives units produced t hours after the factory opened at 8am.At what time during the day is factory productivity a maximum? Answer in the form "_:_ _" (without an "am" or "pm").

Accepted Answer

The answer of Daily production levels in a plant can...

Question 2

If you throw a stone into the air at an angle of $\theta$ to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{100}\left(1+\tan ^{2} \theta\right)$ , where y is the height of the stone above the ground, x is the horizontal distance.If the angle $\theta$ is fixed, what value of x gives the maximum height? (Your answer will $\theta$.)
A) $\frac{50 \tan \theta}{1+\tan ^{2} \theta}$
B) $\frac{1+\tan ^{2} \theta}{50 \tan \theta}$
C) $\frac{\tan \theta}{50+50 \tan ^{2} \theta}$
D) $\frac{50+50 \tan ^{2} \theta}{\tan \theta}$

Accepted Answer

To find the value of $x$ that gives the maximum height, we differentiate the given equation with respect to $x$ and set the derivative equal to zero. The derivative of $y$ with respect to $x$ is $\tan \theta - \frac{2x}{100}(1+\tan^2 \theta)$. Setting this equal to zero and solving for $x$ gives $x = \frac{50 \tan \theta}{1+\tan^2 \theta}$, which is option A.

Question 3

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm³ per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.

Accepted Answer

The answer of A student is drinking a milkshake with...

Question 4

The function $C(r)=10 r^{2}-30$ gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

Accepted Answer

The answer of The function $C(r)=10 r^{2}-30$ gives cost in...

Question 5

A rectangular swimming pool is 10 meters long and 6 meters wide.It has a depth of 1 meter at the shallow end, then slopes to a depth of 1.5 meters at the deep end, as shown in the following cross section (not to scale).It is being filled with a hose at a rate of 50,000 cubic centimeters per minute.225 minutes after the hose is turned on, the water is rising at a rate of _____ cm per second.Round to 3 decimal places.

Accepted Answer

The answer of A rectangular swimming pool is 10 meters...

Question 6

The function $y=0.2(x+2)^{2}-2 x+10$ gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

Accepted Answer

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Question 7

Find the quantity q which maximizes profit if the total revenue, R(q), and the total cost, C(q), are given in dollars by $R(q)=8 q-0.02 q^{2}$ $C(q)=300+1.9 q$ , where $0 \leq q \leq 600$ units.
Round to the nearest whole number.

Accepted Answer

The answer of Find the quantity q which maximizes profit...

Question 8

If you throw a stone into the air at an angle of $\theta$to the horizontal, it moves along the curve $y=x \tan \theta-\frac{x^{2}}{160}\left(1+\tan ^{2} \theta\right)$ ,
where y is the height of the stone above the ground, x is the horizontal distance.Suppose the stone is to be thrown over a wall at a fixed horizontal distance l away from you.If you can vary $\theta$, what is the highest wall that the stone can go over? (Your answer will contain l.)

Accepted Answer

The answer of If you throw a stone into the...

Question 9

What is the shortest distance from the point (0,1)to the curve $y=e^{x}$ ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

Accepted Answer

The answer of What is the shortest distance from the...

Question 10

A fan is watching a 100-meter footrace from a seat in the bleachers 15 meters back from the midway point.The winning runner is moving approximately 8 meters per second.How fast is the distance from the fan to the winning runner changing when he is x meters into the race?

Accepted Answer

The answer of A fan is watching a 100-meter footrace...

Question 11

A cupful of olive oil falls on the floor forming a circular puddle.Its radius is increasing at a constant rate of 0.2 cm/sec.What is the rate of increase in the area of the olive oil when its circumference measures 20 $\pi$ cm?

Accepted Answer

The answer of A cupful of olive oil falls on...

Question 12

A bar of ice cream, with dimensions of 3 cm by 3 cm by 3 cm placed on a mesh screen on top of a cylindrical funnel that is 6 cm high and 6 cm in diameter.If the ice cream is melting at a rate of 3.3 $\mathrm{cm}^{3} / \mathrm{min}$ into the funnel, what is the rate of change of the height of the funnel when half of the ice cream has melted? $V_{f u m a l}=\frac{1}{3} \pi r^{2} h$
A)0.232
B)0.032
C)0.132
D)0.284
E)0.083

Accepted Answer

The volume of the ice cream bar is $3 \times 3 \times 3 = 27 \, \mathrm{cm}^3$. Half melted means $13.5 \, \mathrm{cm}^3$ has melted into the funnel. The volume of a cylinder (funnel in this case) is $V = \pi r^2 h$, but since the funnel is given as a cone shape in the formula, we correct the formula to $V = \frac{1}{3} \pi r^2 h$. The radius $r = 3 \, \mathrm{cm}$ (since diameter is 6 cm). To find the rate of change of height, $dh/dt$, when $13.5 \, \mathrm{cm}^3$ has melted, we use the given melting rate of $3.3 \, \mathrm{cm}^3/\mathrm{min}$. The formula rearranges to $h = \frac{3V}{\pi r^2}$. Differentiating both sides with respect to time $t$ gives $\frac{dh}{dt} = \frac{3}{\pi r^2} \frac{dV}{dt}$. Substituting $r = 3 \, \mathrm{cm}$ and $\frac{dV}{dt} = 3.3 \, \mathrm{cm}^3/\mathrm{min}$ yields $\frac{dh}{dt} = \frac{3 \times 3.3}{\pi \times 3^2} = \frac{9.9}{9\pi} \approx 0.35/\pi \approx 0.111$. The closest answer provided is 0.132, indicating a possible miscalculation in the final step or a rounding difference.

Question 13

A submarine can travel 30mi/hr submerged and 60mi/hr on the surface.The submarine must stay submerged if within 200 miles of shore.Suppose that this submarine wants to meet a surface ship 200 miles off shore.The submarine leaves from a port 300 miles along the coast from the surface ship.What route of the type sketched below should the sub take to minimize its time to rendezvous? Give the value of y to 2 decimal places.

Accepted Answer

The answer of A submarine can travel 30mi/hr submerged and...

Question 14

A spherical lollipop has a circumference of 7.9 centimeters.A student decides to measure the rate of change of the volume of the lollipop, in $\mathrm{cm}^{3}$ per minute.The student licks the lollipop and measures the circumference every minute.The radius is decreasing at a rate of 0.18 cm/min.Determine the rate at which the volume is changing when the circumference is half of it's original size.[ $V=\frac{4}{3} \pi r^{3}$ ]

Accepted Answer

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Question 15

A normal distribution in statistics is modeled by the function $N(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$ determine where the maximum value of the function would occur. A)-2 B)-1 C)0 D)-2 $\pi$ E) $\ell$ ^-²

Accepted Answer

The maximum value of the normal distribution function occurs at $x = 0$, because the exponent $-x^2 / 2$ reaches its maximum value (0) when $x = 0$, making the entire function maximum at this point.

Question 16

Find the marginal cost for q = 100 when the fixed costs in dollars are 1000, the variable costs are $190 per item, and each sells for $310.

Accepted Answer

The answer of Find the marginal cost for q =...

Question 17

The regular air fare between Boston and San Francisco is $600.An airline flying 747s with a capacity of 480 on this route observes that they fly with an average of 400 passengers.Market research tells the airlines' managers that each $20 fare reduction would attract, on average, 20 more passengers for each flight.How should they set the fare to maximize their revenue?

Accepted Answer

The answer of The regular air fare between Boston and...

Question 18

The number of plants in a terrarium is given by the function $P(c)=-1.4 c^{2}+3 c+15$ , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Accepted Answer

The answer of The number of plants in a terrarium...

Question 19

Air is being blown into a spherical balloon at a rate of 70 cm³ per second.At what rate is the surface area of the balloon increasing when the radius is 10 cm? Round to 2 decimal places, and do not include units.

Accepted Answer

The answer of Air is being blown into a spherical...

Question 20

A Brian's candy sugar wand is made from flavored sugar inside a straw.The straw is 210 mm long and 5 mm in diameter.The child accidentally poked a hole in the bottom, making the height of the sugar fall at a rate of 1 mm per second.The child realizes that there is a hole after 1 seconds.What was the rate of change of the volume of the sugar at this time?
[ $V=\pi r^{2} h$ ]

Accepted Answer

The answer of A Brian's candy sugar wand is made...

Quiz 4: Using the Derivative

Daily production levels in a plant can be modeled by the function $G(t)=-3 t^{2}$ $+18t-9$ , which gives units produced t hours after the factory opened at 8am.At what time during the day is factory productivity a maximum? Answer in the form "_:_ _" (without an "am" or "pm").

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm³ per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.

The function $C(r)=10 r^{2}-30$ gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

The function $y=0.2(x+2)^{2}-2 x+10$ gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

Find the quantity q which maximizes profit if the total revenue, R(q), and the total cost, C(q), are given in dollars by $R(q)=8 q-0.02 q^{2}$ $C(q)=300+1.9 q$ , where $0 \leq q \leq 600$ units. Round to the nearest whole number.

What is the shortest distance from the point (0,1)to the curve $y=e^{x}$ ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

A fan is watching a 100-meter footrace from a seat in the bleachers 15 meters back from the midway point.The winning runner is moving approximately 8 meters per second.How fast is the distance from the fan to the winning runner changing when he is x meters into the race?

A cupful of olive oil falls on the floor forming a circular puddle.Its radius is increasing at a constant rate of 0.2 cm/sec.What is the rate of increase in the area of the olive oil when its circumference measures 20 $\pi$ cm?

A normal distribution in statistics is modeled by the function $N(x) =\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$ determine where the maximum value of the function would occur.

Find the marginal cost for q = 100 when the fixed costs in dollars are 1000, the variable costs are $190 per item, and each sells for $310.

The number of plants in a terrarium is given by the function $P(c)=-1.4 c^{2}+3 c+15$ , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Air is being blown into a spherical balloon at a rate of 70 cm³ per second.At what rate is the surface area of the balloon increasing when the radius is 10 cm? Round to 2 decimal places, and do not include units.

Quiz 4: Using the Derivative

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm3 per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.

The function C(r)=10r2−30C(r)=10 r^{2}-30C(r)=10r2−30 gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

The function y=0.2(x+2)2−2x+10y=0.2(x+2)^{2}-2 x+10y=0.2(x+2)2−2x+10 gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

What is the shortest distance from the point (0,1)to the curve y=exy=e^{x}y=ex ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

A fan is watching a 100-meter footrace from a seat in the bleachers 15 meters back from the midway point.The winning runner is moving approximately 8 meters per second.How fast is the distance from the fan to the winning runner changing when he is x meters into the race?

A cupful of olive oil falls on the floor forming a circular puddle.Its radius is increasing at a constant rate of 0.2 cm/sec.What is the rate of increase in the area of the olive oil when its circumference measures 20 π\piπ cm?

A normal distribution in statistics is modeled by the function N(x) =12πe−x2/2N(x) =\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}N(x) =2π​1​e−x2/2 determine where the maximum value of the function would occur.

Find the marginal cost for q = 100 when the fixed costs in dollars are 1000, the variable costs are $190 per item, and each sells for $310.

The number of plants in a terrarium is given by the function P(c)=−1.4c2+3c+15P(c)=-1.4 c^{2}+3 c+15P(c)=−1.4c2+3c+15 , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Air is being blown into a spherical balloon at a rate of 70 cm3 per second.At what rate is the surface area of the balloon increasing when the radius is 10 cm? Round to 2 decimal places, and do not include units.

A student is drinking a milkshake with a straw from a cylindrical cup with a radius of 5.5 cm.If the student is drinking at a rate of 4.5 cm³ per second, then the level of the milkshake dropping at a rate of _____ cm per second.Round to 2 decimal places.

The function $C(r)=10 r^{2}-30$ gives cost in dollars of producing r items.What is the marginal cost of increasing r by 1 item from the current production level of r = 6?

The function $y=0.2(x+2)^{2}-2 x+10$ gives the population of a town (in 1000's of people)at time x where x is the number of years since 1980.When was the population a minimum? Round to the nearest year.

What is the shortest distance from the point (0,1)to the curve $y=e^{x}$ ? You will need to use a calculator with root-finding capabilities.Give your answer to 2 decimal places.

A cupful of olive oil falls on the floor forming a circular puddle.Its radius is increasing at a constant rate of 0.2 cm/sec.What is the rate of increase in the area of the olive oil when its circumference measures 20 $\pi$ cm?

A normal distribution in statistics is modeled by the function $N(x) =\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$ determine where the maximum value of the function would occur.

The number of plants in a terrarium is given by the function $P(c)=-1.4 c^{2}+3 c+15$ , where c is the number of mg of plant food added to the terrarium.Find the amount of plant food that produces the highest number of plants.Round to 2 decimal places.

Air is being blown into a spherical balloon at a rate of 70 cm³ per second.At what rate is the surface area of the balloon increasing when the radius is 10 cm? Round to 2 decimal places, and do not include units.