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Calculus 1 : Rate

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Example Questions

Example Question #41 : Rate

If water is poured inside a cylindrical cup with radius of 3 inches at a rate of 2 cubic inches per second, how fast is the surface rising?

Possible Answers:

$\frac{3}{2\pi}$

$\frac{2}{9\pi}$

$\frac{9\pi}{2}$

$\frac{2\pi}{3}$

$\frac{6\pi}{5}$

Correct answer:

$\frac{2}{9\pi}$

Explanation:

Write the volume of the cylinder.

$V=\pi r^2 h$

Substitute the radius.

$V=\pi (3)^2 h= 9\pi h$

Differentiate the volume with respect to time.

$\frac{dV}{dt}=9\pi \frac{dh}{dt}$

Substitute the given rate.

$2=9\pi \frac{dh}{dt}$

$\frac{dh}{dt}=\frac{2}{9\pi}$

The height is increasing at a rate of $\frac{2}{9\pi}$ inches per second.

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Example Question #42 : Rate

The initial position of a particle is 0. If its velocity is described by v(t) = 4t + 5, what is its position at the time when the velocity is equal to 45?

Possible Answers:

250

173

105

125

Correct answer:

250

Explanation:

The position s(t) is equal to ∫v(t) dt = ∫ 4t + 5 dt = 2t² + 5t + C

Now, since we know that the initial position of the particle (at t = 0) is 0, we know C is 0. Therefore, s(t) = 2t² + 5t + C

To find the time t when at which the velocity is 45, set v(t) equal to 45. 45 = 4t + 5 → 40 = 4t → t = 10

The position of the particle is s(10) = 2 * 10² + 50 = 200 + 50 = 250

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Example Question #43 : Rate

The position of a particle is defined by s(t) = 4t³ – 3t² + 2t. At what time (to the nearest hundreth) is its velocity equal to 384?

Possible Answers:

(22 ± √(5))/10

48 ± √(17)

15 ± √(3)

(3 ± √(4593))/12

(4 ± √(4721))/2

Correct answer:

(3 ± √(4593))/12

Explanation:

The position of a particle is defined by s(t) = 4t³ – 3t² + 2t. At what time (to the nearest hundreth) is its velocity equal to 384?

Start by finding the velocity function:

v(t) = s'(t) = 12t² – 6t + 2

To find the time, set v(t) equal to 384: 384 = 12t² – 6t + 2

To solve, set equal the equation equal to 0: 12t² – 6t – 382 = 0

Use the quadratic formula:

(–(–6) ± √(36 – 4*12*(–382)))/(2 * 12)

= (6 ± √(36 + 18336))/24 = (6 ± √(18372))/24 = (6 ± 2√(4593))/24

= (3 ± √(4593))/12

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Example Question #44 : Rate

The initial position of a particle is 44.5. If its velocity is described by v(t) = 3t + 12, what is its position at the time when the velocity is equal to 8391?

Possible Answers:

11734834

11524231

1038181

11734745

11734789.5

Correct answer:

11734834

Explanation:

The initial position of a particle is 44.5. If its velocity is described by v(t) = 3t + 12, what is its position (to the nearest hundreth) at the time when the velocity is equal to 8391?

To solve for t, set v(t) equal to 8391: 3t + 12 = 8391; 3t = 8379; t = 2793

Now, the position function is equal to ∫v(t)dt = ∫ 3t + 12 dt = (3/2)t² + 12t + C

Now, since the initial position is 44.5, C is 44.5; therefore, s(t) = (3/2)t² + 12t + 44.5

To solve for the position, solve s(2793) = (3/2)2793² + 12*2793 + 44.5 = 1.5 * 7800849 + 33516 + 44.5 = 11701273.5 + 33516 + 44.5

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Example Question #45 : Rate

An eagle flies at a parabolic trajectory such that h(t)= 25-2t^2 , where h(t) is in the height in meters and is the time in seconds. At what time will its velocity $v=-8 \frac{m}{s}$ ?

Possible Answers:

6.3

$\sqrt8.5$

Correct answer:

Explanation:

Take the derivative of the position function to obtain the velocity function.

$v(t)=\frac{\mathrm{d} }{\mathrm{d} t}h(t)=-4t$

We want to know the time when the velocity is -8. Substitute v into the equation to find t.

-8=-4t

2=t

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Example Question #3 : How To Solve For Time

A ball is thrown in the air, modeled by the function y(t)=-2t^2+t+3 . At what time will the ball hit the ground?

Possible Answers:

$t=\frac{3}{2}$

t=2

$t=\frac{1}{2}$

t=0

$t=\frac{1}{4}$

Correct answer:

$t=\frac{3}{2}$

Explanation:

To find the time when the ball hits the ground, set y(t)=0 and solve for .

0=-2t^2+t+3

0=-(2t^2-t-3)

0=(2t-3)(t+1)

Separate each term and solve for t.

2t-3=0

$t=\frac{3}{2}$

t+1=0

t=-1

Since negative time does not exist, the only possible answer is $t=\frac{3}{2}$ .

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Example Question #3 : How To Solve For Time

Suppose the acceleration function of a biker going uphill at the start of a race is a(t)= 2t-8 , where is in seconds. When will it take the biker to reach constant velocity?

Possible Answers:

$2\sqrt2$

Correct answer:

Explanation:

Constant velocity means there is neither an increase or decrease in acceleration.

a=0

Substitute acceleration and solve for time.

a(t)= 2t-8

0=2t-8

t=4

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Example Question #4 : How To Solve For Time

For this problem, the acceleration of gravity is simplified to $10 \frac{m}{s^2}$ .

I fire a cannon straight into the air. Assuming that the cannonball leaves the cannon at a velocity of $100 \frac{m}{s}$ , weighs kgs, and is fired from the ground (i.e. m), how long will it take before the cannonball reaches the ground again?

Possible Answers:

$10\ seconds$

$-10\ seconds$

$100 \frac{m}{s}$

$5\ seconds$

$20\ seconds$

Correct answer:

$20\ seconds$

Explanation:

Initial velocity is given as 100 m/s and the acceleration due to gravity is $10\frac{m}{s^2}$ towards the Earth , or $-10 \frac{m}{s^2}$ .

We want to find how long it will take for the velocity of the cannonball to reach , so we set $v = 100 + a\cdot t = 0$ , where t is time and $a = -10 \frac{m}{s^2}$ .

So, 100 -10t = 0 , 100 = 10t , t = 10 .

Therefore, it takes ten seconds for the ball to reach a velocity of 0, and given that acceleration is uniform (i.e. a constant), we know that it will take the same amount of time to come down as it took to go up, or ten seconds. Therefore, the total time the cannonball spends in the air is,

10+10 = 20 seconds.

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Example Question #3 : How To Solve For Time

Given the instantaneous velocity and the position function, find the time at which the moving object reaches that instantaneous velocity.

v=125, p(t)=12t^2+29t-34

Possible Answers:

$t=\frac{1}{4}$

t=2.63

t=-5.04

t=4

Correct answer:

t=4

Explanation:

We begin by finding the derivative of the position function using the power rule:

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

p'(t)=v(t)=24t+29

We then set the given instantaneous velocity equal to the velocity function and solve for t:

$24t+29=125\Rightarrow t=4$

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Example Question #3 : How To Solve For Time

Assume the acceleration due to gravity is -9.8 m/s^2

If you throw a ball straight up with an initial velocity of 9.8 m/s how many seconds will it take before the ball returns to your hand?

Possible Answers:

0.5

1.5

0.64

Correct answer:

Explanation:

Write out the equation for the height of the ball

$h=-\frac{9.8}{2}t^{2}+9.8t$ .

You can arrive at this by starting with the information that the acceleration on the ball is the constant acceleration due to gravity and integrating twice. Making sure to solve for your constants along the way.

$v(t)=\int a(t)dt = \int -9.8 dt = -9.8t+v_{0}$

$x(t)=\int v(t)dt =\int -9.8t+v_{0}dt=-\frac{9.8}{2}t^{2}+v_{0}t+x_{0}$

Your initial velocity of 9.8m/s and your initial position of will help write out this equation.

$x(t)=-\frac{9.8}{2}t^2+9.8t+0$

Then solve for the two values of for which the ball is at height . Those are seconds and seconds.

$\\x(t)=-\frac{9.8}{2}t^2+9.8t+0=0\\ \\ -4.9t^{2}+9.8t=0\\ \\ t(-4.9t+9.8)=0\\ \\t=0,t=2$

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Calculus 1 : Rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #41 : Rate

Example Question #42 : Rate

Example Question #43 : Rate

Example Question #44 : Rate

Example Question #45 : Rate

Example Question #3 : How To Solve For Time

Example Question #3 : How To Solve For Time

Example Question #4 : How To Solve For Time

Example Question #3 : How To Solve For Time

Example Question #3 : How To Solve For Time

All Calculus 1 Resources

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