Chapter 4

Find the quantities for the given equation. Find \(\frac{d y}{d t}\) at \(x=1\) and \(y=x^{2}+3\) if \(\frac{d x}{d t}=4\)

Find the quantities for the given equation. Find \(\frac{d y}{d t}\) at \(x=1\) and \(y=x^{2}+3\) if \(\frac{d x}{d t}=4\).

Find the quantities for the given equation. Find \(\frac{d x}{d t}\) at \(x=-2\) and \(y=2 x^{2}+1\) if \(\frac{d y}{d t}=-1.\)

Find the quantities for the given equation. Find \(\frac{d x}{d t}\) at \(x=-2\) and \(y=2 x^{2}+1\) if \(\frac{d y}{d t}=-1\).

Find the quantities for the given equation. Find \(\frac{d z}{d t}\) at \((x, y)=(1,3)\) and \(z^{2}=x^{2}+y^{2}\) if \(\frac{d x}{d t}=4\) and \(\frac{d y}{d t}=3.\)

Find the quantities for the given equation. Find \(\frac{d z}{d t}\) at \((x, y)=(1,3)\) and \(z^{2}=x^{2}+y^{2}\) if \(\frac{d x}{d t}=4\) and \(\frac{d y}{d t}=3 .\)

Sketch the situation if necessary and used related rates to solve for the quantities. If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, \(\Omega\) ) is given by the equation \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} .\) If \(R_{1}\) is increasing at a rate of 0.5\(\Omega / \min\) and \(R_{2}\) decreases at a rate of 1.1\(\Omega / \mathrm{min}\) , at what rate does the total resistance change when \(R_{1}=20 \Omega\) and \(R_{2}=50 \Omega / \min ?\)

Sketch the situation if necessary and used related rates to solve for the quantities. A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?

Sketch the situation if necessary and used related rates to solve for the quantities. A 25-ft ladder is leaning against a wall. If we push the ladder toward the wall at a rate of 1 ft/sec, and the bottom of the ladder is initially 20 ft away from the wall, how fast does the ladder move up the wall 5 sec after we start pushing?

Sketch the situation if necessary and used related rates to solve for the quantities. Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at 300 mi/h. If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B, at what rate is the distance between the airplanes changing?

Sketch the situation if necessary and used related rates to solve for the quantities. You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding 12 mph north. After you traveled 4 mi, at what rate is the distance between you changing?

Sketch the situation if necessary and used related rates to solve for the quantities. Two buses are driving along parallel freeways that are 5 mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55 mph, find the rate at which the distance between the buses is changing when they are 13 mi apart, heading toward each other.

Sketch the situation if necessary and used related rates to solve for the quantities. A 6-ft-tall person walks away from a 10-ft lamppost at a constant rate of 3 ft/sec. What is the rate that the tip of the shadow moves away from the pole when the person is 10 ft away from the pole?

Sketch the situation if necessary and used related rates to solve for the quantities. A 5-ft-tall person walks toward a wall at a rate of 2 ft/ sec. A spotlight is located on the ground 40 ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is 10 ft from the wall?

Sketch the situation if necessary and used related rates to solve for the quantities. A helicopter starting on the ground is rising directly into the air at a rate of 25 ft/sec. You are running on the ground starting directly under the helicopter at a rate of 10 ft/sec. Find the rate of change of the distance between the helicopter and yourself after 5 sec.

For the following exercises, draw and label diagrams to help solve the related-rates problems. The side of a cube increases at a rate of \(\frac{1}{2} \mathrm{m} / \mathrm{sec} .\) Find the rate at which the volume of the cube increases when the side of the cube is 4 m.

Draw and label diagrams to help solve the related-rates problems. The side of a cube increases at a rate of \(\frac{1}{2} \mathrm{~m} / \mathrm{sec}\). Find the rate at which the volume of the cube increases when the side of the cube is \(4 \mathrm{~m}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. The volume of a cube decreases at a rate of 10 m/sec. Find the rate at which the side of the cube changes when the side of the cube is 2 m.

Draw and label diagrams to help solve the related-rates problems. The volume of a cube decreases at a rate of \(10 \mathrm{~m} / \mathrm{sec}\). Find the rate at which the side of the cube changes when the side of the cube is \(2 \mathrm{~m}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. The radius of a circle increases at a rate of 2 m/sec. Find the rate at which the area of the circle increases when the radius is 5 m.

Draw and label diagrams to help solve the related-rates problems. The radius of a circle increases at a rate of \(2 \mathrm{~m} / \mathrm{sec}\). Find the rate at which the area of the circle increases when the radius is \(5 \mathrm{~m}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. The radius of a sphere decreases at a rate of 3 m/sec. Find the rate at which the surface area decreases when the radius is 10 m.

Draw and label diagrams to help solve the related-rates problems. The radius of a sphere decreases at a rate of \(3 \mathrm{~m} / \mathrm{sec}\). Find the rate at which the surface area decreases when the radius is \(10 \mathrm{~m}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. The radius of a sphere increases at a rate of 1 m/sec. Find the rate at which the volume increases when the radius is 20 m.

Draw and label diagrams to help solve the related-rates problems. The radius of a sphere increases at a rate of \(1 \mathrm{~m} / \mathrm{sec}\). Find the rate at which the volume increases when the radius is \(20 \mathrm{~m}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. The radius of a sphere is increasing at a rate of 9 cm/ sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

Draw and label diagrams to help solve the related-rates problems. The radius of a sphere is increasing at a rate of \(9 \mathrm{~cm} /\) sec. Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.

For the following exercises, draw and label diagrams to help solve the related-rates problems. The base of a triangle is shrinking at a rate of 1 cm/min and the height of the triangle is increasing at a rate of 5 cm/min. Find the rate at which the area of the triangle changes when the height is 22 cm and the base is 10 cm.

Draw and label diagrams to help solve the related-rates problems. The base of a triangle is shrinking at a rate of \(1 \mathrm{~cm} / \mathrm{min}\) and the height of the triangle is increasing at a rate of \(5 \mathrm{~cm} /\) min. Find the rate at which the area of the triangle changes when the height is \(22 \mathrm{~cm}\) and the base is \(10 \mathrm{~cm}\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. A triangle has two constant sides of length 3 ft and 5 ft. The angle between these two sides is increasing at a rate of 0.1 rad/sec. Find the rate at which the area of the triangle is changing when the angle between the two sides is \(\pi / 6\).

Draw and label diagrams to help solve the related-rates problems. A triangle has two constant sides of length \(3 \mathrm{ft}\) and 5 ft. The angle between these two sides is increasing at a rate of \(0.1 \mathrm{rad} / \mathrm{sec}\). Find the rate at which the area of the triangle is changing when the angle between the two sides is \(\pi / 6\).

For the following exercises, draw and label diagrams to help solve the related-rates problems. A triangle has a height that is increasing at a rate of 2 \(\mathrm{cm} /\) sec and its area is increasing at a rate of 4 \(\mathrm{cm}^{2} / \mathrm{sec}\) . Find the rate at which the base of the triangle is changing when the height of the triangle is 4 \(\mathrm{cm}\) and the area is 20 \(\mathrm{cm}^{2} .\)

Draw and label diagrams to help solve the related-rates problems. A triangle has a height that is increasing at a rate of 2 \(\mathrm{cm} / \mathrm{sec}\) and its area is increasing at a rate of \(4 \mathrm{~cm}^{2} / \mathrm{sec} .\) Find the rate at which the base of the triangle is changing when the height of the triangle is \(4 \mathrm{~cm}\) and the area is \(20 \mathrm{~cm}^{2}\).

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. How fast does the depth of the water change when the water is 10 \(\mathrm{ft}\) high if the cone leaks water at a rate of 10 \(\mathrm{ft}^{3} / \mathrm{min} ?\)

Consider a right cone that is leaking water. The dimensions of the conical tank are a height of \(16 \mathrm{ft}\) and a radius of \(5 \mathrm{ft}\). How fast does the depth of the water change when the water is \(10 \mathrm{ft}\) high if the cone leaks water at a rate of 10 \(\mathrm{ft}^{3} / \mathrm{min} ?\)

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. Find the rate at which the surface area of the water changes when the water is 10 \(\mathrm{ft}\) high if the cone leaks water at a rate of 10 \(\mathrm{ft}^{3} / \mathrm{min.}\)

Consider a right cone that is leaking water. The dimensions of the conical tank are a height of \(16 \mathrm{ft}\) and a radius of \(5 \mathrm{ft}\). Find the rate at which the surface area of the water changes when the water is \(10 \mathrm{ft}\) high if the cone leaks water at a rate of \(10 \mathrm{ft}^{3} / \mathrm{min}\).

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. If the water level is decreasing at a rate of 3 in./min when the depth of the water is 8 \(\mathrm{ft}\) , determine the rate at which water is leaking out of the cone.

Consider a right cone that is leaking water. The dimensions of the conical tank are a height of \(16 \mathrm{ft}\) and a radius of \(5 \mathrm{ft}\). If the water level is decreasing at a rate of 3 in./min when the depth of the water is \(8 \mathrm{ft}\), determine the rate at which water is leaking out of the cone.

A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of \(2 \mathrm{~m}\) and a radius of \(2 \mathrm{~m}\). Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is \(10 \mathrm{~cm} / \mathrm{min}\) when the height is \(1 \mathrm{~m}\).

For the following exercises, consider a right cone that is leaking water. The dimensions of the conical tank are a height of 16 ft and a radius of 5 ft. A trough has ends shaped like isosceles triangles, with width 3 m and height 4 m, and the trough is 10 m long. Water is being pumped into the trough at a rate of 5 \(\mathrm{m}^{3} / \mathrm{min}\) . At what rate does the height of the water change when the water is 1 \(\mathrm{m}\) deep?

Consider a right cone that is leaking water. The dimensions of the conical tank are a height of \(16 \mathrm{ft}\) and a radius of \(5 \mathrm{ft}\). A trough has ends shaped like isosceles triangles, with width \(3 \mathrm{~m}\) and height \(4 \mathrm{~m}\), and the trough is 10 \(\mathrm{m}\) long. Water is being pumped into the trough at a rate of \(5 \mathrm{~m}^{3} / \mathrm{min}\). At what rate does the height of the water change when the water is \(1 \mathrm{~m}\) deep?

For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\) . The radius of the pool is 10 \(\mathrm{ft}\) . Find the rate at which the depth of the water is changing when the water has a depth of 1 ft.

Consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\). The radius of the pool is \(10 \mathrm{ft}\). Find the rate at which the depth of the water is changing when the water has a depth of \(5 \mathrm{ft}\).

Consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\). The radius of the pool is \(10 \mathrm{ft}\). Find the rate at which the depth of the water is changing when the water has a depth of \(1 \mathrm{ft}\).

For the following problems, consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\) . The radius of the pool is 10 \(\mathrm{ft}\) . Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of 10 \(\mathrm{ft}^{3} / \mathrm{min}\) . The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of 5 \(\mathrm{ft.}\)

Consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\). The radius of the pool is \(10 \mathrm{ft}\). Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of \(10 \mathrm{ft}^{3} / \mathrm{min}\). The radius of the cone base is three times the height of the cone. Find the rate at which the height of the gravel changes when the pile has a height of \(5 \mathrm{ft}\).

For the following exercises, draw the situations and solve the related-rate problems. You are stationary on the ground and are watching a bird fly horizontally at a rate of 10 m/sec. The bird is located 40 m above your head. How fast does the angle of elevation change when the horizontal distance between you and the bird is 9 m?

For the following exercises, draw the situations and solve the related-rate problems. You stand 40 ft from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of 20 ft/sec. Find the rate at which the angle of elevation changes when the rocket is 30 ft in the air.

Consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\). The radius of the pool is \(10 \mathrm{ft}\). You stand \(40 \mathrm{ft}\) from a bottle rocket on the ground and watch as it takes off vertically into the air at a rate of \(20 \mathrm{ft} /\) sec. Find the rate at which the angle of elevation changes when the rocket is \(30 \mathrm{ft}\) in the air.