Review #4 - Related Rates & Riemann Sums

Situation for #1 - #7:
Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate is given in the problem?

•  $\frac{dr}{dt}=40$ 

•  $\frac{dC}{dt}=40$ 

•  $\frac{dA}{dt}=40$ 

•  $\frac{dπ}{dt}=40$ 

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What rate are we looking for and when?

•  $\frac{dA}{dt}\ when\ A=100π$ 

•  $\frac{dC}{dt}\ when\ A\ =\ 100π$ 

•  $\frac{dA}{dt}\ when\ C\ =\ 100π$ 

•  $\frac{dC}{dt}\ when\ r\ =\ 100π$ 

What is the derivative of area with respect to time?

• $\frac{dA}{dt}=2πr\cdot\frac{dr}{dt}$

• $\frac{dA}{dt}=2πr$

• $\frac{dA}{dt}=πr\cdot\frac{dr}{dt}$

• $\frac{dA}{dt}=πr^2\cdot\frac{dr}{dt}$

Tags

CCSS.HSF.BF.A.1

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

What is the length of the radius when the circumference is 100π ft?

•  $r=50\ ft$ 

•  $r=100\ ft$ 

•  $r=\sqrt{50}ft$ 

• cannot be determined

Tags

CCSS.HSA.SSE.A.1

CCSS.HSG.GMD.A.1

What is the derivative of circumference with respect to time?

• $\frac{dC}{dt}=π\cdot\frac{dr}{dt}$

• $\frac{dC}{dt}=2π$

• $\frac{dC}{dt}=2πr$

• $\frac{dC}{dt}=2π\cdot\frac{dr}{dt}$

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

At what rate is the radius changing when the circumference is 100π ft?

•  $\frac{dr}{dt}=\frac{50}{π}$ 

•  $\frac{dr}{dt}=\frac{20}{π}$ 

•  $\frac{dr}{dt}=50$ 

•  $\frac{dr}{dt}=20$  

Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100π ft?

•  $\frac{dA}{dt}=2000\ \frac{ft^2}{\sec}$ 

•  $\frac{dA}{dt}=40\ \frac{ft^2}{\sec}$ 

•  $\frac{dA}{dt}=100\ \frac{ft^2}{\sec}$ 

•  $\frac{dA}{dt}=200\ \frac{ft^2}{\sec}$  

Tags

CCSS.HSA.CED.A.2

CCSS.HSA.SSE.A.1

CCSS.HSG.GMD.A.1

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

• 3 ft/sec
• -7.2 ft/sec
• 7.2 ft/sec
• 12

Tags

CCSS.HSA.CED.A.2

CCSS.HSG.SRT.C.6

CCSS.HSG.SRT.C.8

A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 4 in? (Volume of a sphere =  $\frac{4}{3}\pi r^3$  )

• -262π in³/sec

• -247π in³/sec

• -256π in³/sec

• -263π in³/sec

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is full and begins draining at a rate of 1 ft³/sec. How fast is the height of the water changing when the water is 2 ft deep?

To answer this question, you would need to write the volume of the cone in terms of height before differentiating.  What could you replace the radius with?

• $r\ =\ h^2$

• $r\ =\ 4h$

• $r\ =\ \frac{2}{3}h$

• $r\ =\ \frac{h}{2}$