4.6: Related Rates

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Mar 12, 2024
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- 4.5: Limits at Infinity and Asymptotes
- 4.7: L'Hôpital's Rule

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A $10$ -ft ladder is leaning lagainst 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?
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. 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?
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?

$A lamppost is shown that is 10 ft high. To its right, there is a person who is 6 ft tall. There is a line from the top of the lamppost that touches the top of the person’s head and then continues to the ground. The length from the end of this line to where the lamppost touches the ground is 10 + x. The distance from the lamppost to the person on the ground is 10, and the distance from the person to the end of the line is x.$
A $5$ -ft-tall person walks toward a wall at a rate of $2$ ft/sec. A spotlight located on the ground is $40$ 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?
The side of a cube increases at a rate of $\dfrac{1}{2}$ m/sec. Find the rate at which the volume of the cube increases when the side of the cube is $4$ m.
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.
A cylindrical tank standing upright (with one circular base on the ground) has a radius of $20$ cm. How fast does the water level in the tank drop when the water is being drained at $25$ cm³/sec?
A tank in the shape of an inverted right cone (that is, a cone with the point facing down, like a funnel) is leaking water at a rate of $10$ ft³/min. The tank is $16$ ft high and has a radius of $5$ ft. How fast does the depth of the water change when the water is $10$ ft high?
Sand is poured onto a surface at a rate of $15$ cm³/sec, forming a conical pile whose base diameter is always equal to its height. How fast is the height of the pile increasing when the pile is $3$ cm high?
A baseball diamond is a square with each side measuring $90$ ft. A player runs from first base to second base at $15$ ft/sec. At what rate is the player's distance from third base decreasing when they are half way from first to second base?

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