[Notes & HW Answers] 4.6 Related Rates

[Notes & HW Answers] 4.6 Rates and Related Rates.pdf

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[Prepwork 4.6]

Q1. In example 4, how fast is the fuel consumption changing if instead the car is moving 30 mph and decelerating at a rate of 3000 miles/hr^2?

Answer: Decreasing at a rate of approximately (choose the closest answer) 600 mpg/hr.

Q2. A spherical snowball is melting. Its radius decreases at a constant rate of 3 cm per minute from an initial value of 100 cm. How fast is the volume decreasing 10 minutes later? Give your answer in cubic centimeters per minute.

A:

Rate = 58800π cubic centimeters per minute.

Q3. An airplane, flying at 500 km/hr at a constant altitude of 3 km is approaching a camera mounted on the ground. Let 𝜃 be the angle of elevation above the ground at which the camera is pointed. When this angle is equal to 𝜋/4, how many radians per hour is the rate of change for the camera's angle? Give your answer in radians/hr.

A:

𝑑𝜃/𝑑𝑡= 500/6​ radians per hour.

[HW 4.6]

Q1. If 𝑥^2+𝑦^2=36 and 𝑑𝑥/𝑑𝑡=6, find 𝑑𝑦/𝑑𝑡 when 𝑦 is positive and

A:

(a) 𝑥=0: 𝑑𝑦/𝑑𝑡= 0

(b) 𝑥=2: 𝑑𝑦/𝑑𝑡=−3√2^−1

(c) 𝑥=3: 𝑑𝑦/𝑑𝑡=−6√3^−1\

Q2. A rectangle has one side of 99 cm. How fast is the area of the rectangle changing at the instant when the other side is 1818 cm and increasing at 22 cm per minute? 

Answer: 18 cm^2/min

Q3. A potter forms a piece of clay into a cylinder. As he rolls it, the length, 𝐿, of the cylinder increases and the radius, 𝑟, decreases. If the length of the cylinder is increasing by 0.4 cm per second, find the rate at which the radius is changing when the radius is 2 cm and the length is 7 cm. 

A: rate = -0.8/14 cm/s