AP Calculus Related Rates

Let x and y be functions of time t such that the sum of x and y is constant. Which of the following equations describes the relationship between the rate of change of x with respect to time and the rate of change of y with respect to time?

• $\frac{\text{d}x}{\text{d}t}=\frac{\text{d}y}{\text{d}t}$

• $\frac{\text{d}x}{\text{d}t}=-\frac{\text{d}y}{\text{d}t}$

• $\frac{\text{d}x}{\text{d}t}+\frac{\text{d}y}{\text{d}t}=\frac{\text{d}K}{\text{d}t}$ , where K is a function of t

• $\frac{\text{d}x}{\text{d}t}+\frac{\text{d}y}{\text{d}t}=K$ , where K is a function of t

Tags

CCSS.HSA.CED.A.2

An isosceles right triangle with legs of length s has area  $A=\frac{1}{2}s^2.$  At the instant when  $s=\sqrt{32}$  centimeters, the area of the triangle is increasing at a rate of 12 square centimeters per second.  At what rate is the length of the hypotenuse of the triangle increasing, in centimeters per second, at that instant?

•  $\frac{3}{4}$  

• 3

•  $\sqrt{32}$  

• 48

Tags

CCSS.HSG.SRT.C.8

CCSS.HSG.SRT.D.11

A tube is being stretched while maintaining its cylindrical shape. The height is increasing at a rate of 2 mm per second. At the instant that the radius of the tube is 6 mm, the volume is increasing at the rate of  $96\pi$  cubic mm per second.  Which of the following statements about the surface area of the tube is true at this instant?  (The Volume V of a cylinder with radius r and height h is  $V=\pi r^2h.$  The surface area S of a cylinder, not including the top and bottom of the cylinder, is  $S=2\pi rh$  )

• The surface area is increasing by  $28\pi$  square mm per second

• The surface area is decreasing by  $28\pi$  square mm per second

• The surface area is increasing by  $32\pi$  square mm per second

• The surface area is decreasing by  $32\pi$  square mm per second

Tags

CCSS.HSG.GMD.A.1

The figure shows an above ground swimming pool in the shape of a cylinder with a radius of 2 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time t=0. During the time interval  $0\le x\le12$  hours, water is pumped into the pool at the rate P(t) cubic feet per hour.  The table gives values of P(t) for selected values of t.  During the same time interval, water is leaking from the pool at the rate R(t) cubic feet per hour, where  $R\left(t\right)=25e^{-0.05t}$  .  Find the rate at which the volume of water in the pool is increasing at time t=8 hours.  How fast is the water level in the pool rising at t= 8 hours?  Indicate units of measure in both answers.  

Tags

CCSS.HSA.CED.A.1

CCSS.HSA.CED.A.2

CCSS.HSA.SSE.A.1

A constant volume of pizza dough is formed into a cylinder with a relatively small height and large radius. The dough is spun and tossed into the air in such a way that the height of the dough decreases as the radius increases, but it retains its cylindrical shape. At time t = k, the height of the dough is 1/3 inch, the radius of the dough is 12 inches, and the radius of the dough is increasing at a rate of 2 inches per minute.

a.) At time t = k, at what rate is the area of the circular surface of the dough increasing with respect to time?

b.) At time t=k, at what rate is the height of the dough decreasing with respect to time?

Tags

CCSS.HSA.CED.A.2

CCSS.HSA.SSE.A.1

CCSS.HSG.GMD.A.3