Edfinity HW 3.11 Related Rates

Air is being pumped into a spherical balloon so that its volume increases at a rate of 90 cm³/s.

How fast is the surface area of the balloon increasing when its radius is 8 cm?

dS/dt = 22.5 cm²/s EXPLANATION:

The top of a 21 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 3 feet per second. How fast is the bottom of the ladder sliding along the ground when the bottom of the ladder is 12 feet away from the base of the wall?

Rate: square root (297) / 4 EXPLANATION:

Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 3 m/s, how fast is the area of the spill increasing when the radius is 35 m?

dA/dt = 210(pi) m²/s EXPLANATION: A = (pi)r² dA/dt = 2(pi)r (dr/dt) Substitute the given values into the differentiated equation. The radius is r = 35 m and the rate of change of the radius is dr/dt = 3 m/s. dA/dt = 2(pi)(35)(3) =659.734 210(pi) m²/s

A conical water tank with vertex down has a radius of 11 feet at the top and is 21 feet high. If water flows into the tank at a rate of 20 ft³/min, how fast is the depth of the water increasing when the water is 17 feet deep?

If the radius of a sphere is increasing at a constant rate of 2 cm/sec, then the volume is increasing at a rate of ____ cm³/sec when the radius is 4 cm.

Suppose that water is pouring into a swimming pool in the shape of a right circular cylinder at a constant rate of 3 cubic feet per minute.

If the pool has radius 3 feet and height 6 feet, what is the rate of change of the height of the water in the pool when the depth of the water in the pool is 2 feet?

A rock is thrown into the reflection pond and causes a circular ripple.

If the radius of the ripple is increasing at a rate of 2 feet per second, how fast is the circumference changing when the radius is 18 feet?

The radius of a spherical balloon is increasing at a rate of 3 centimeters per minute. How fast is the surface area changing when the radius is 8 centimeters?

A Boeing 787 is climbing at a 30 degree angle ((pi)/6 radians) relative to the horizontal.

How fast is the aircraft gaining altitude if its speed is 350 miles/hour?

The height of a right triangle is increasing at a rate of 1 centimeters/minute while the area of the triangle is increasing at a rate of 4 square centimeters/minute.

At what rate is the base of the triangle changing when the height is 7 centimeters and the area is 81 square centimeters?

Gravel is being dumped from a conveyor belt at a rate of 10 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.

How fast is the height of the pile increasing when the pile is 24 feet high?

A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.1 cm/min.

At what rate is the volume of the snowball changing when the diameter is 8 cm?

Two cars start moving from the same point. One travels south at 60 miles/hour and the other travels west at 25 miles/hour.

How fast is the distance between the cars increasing two hours later?