Date modified: Wed 2024-07-31 . 12:10 PM
Date created: Wed 2024-07-31 . 12:10 PM
# Week 8 Problem Set B. ## Reading. Thomas Ch. 3.8 on related rates. Remember to: (1) Draw a diagram, (2) give names to variables and label them, (3) identify the numerical quantities given, and what we want to solve, (4) write down a relation between the variables, (5) implicit differentiate to get a relation between the rates, (6) evaluate! Also review the examples done in class as well! You can do it! ## Problems. 1. If $x^{2}+y^{2}=25$, and $dx / dt = -2$, then what is $dy /dt$ when $x=3$ and $y=-4$? (Assuming $x,y$ are functions of $t$) 2. **A sliding ladder.** A 13-ft ladder is leaning against a wall when its base starts to slide away (see following diagram). By the moment the base is 12ft from the wall, the base is moving at a rate of 5 ft/sec. 1. How fast is the top of the ladder sliding down the wall? 2. At what rate is the area of the triangle formed by the ladder, the wall, and the ground changing? 3. At what rate is the angle $\theta$ between the ladder and the ground changing?![[1 teaching/smc-fall-2023-math-7/week-8/---files/Pasted image 20231019183457.png]] 3. **Flying a kite.** A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft away from her (direct distance)? 4. **A growing sand pile.** Sand falls from a conveyor belt at a rate of $10$ $\text{m}^{3} /\text{min}$ onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. When the pile is $4$ m high, **(a)** how fast is the height changing? **(b)** How fast is the radius changing? Answer in centimeters per minute. 5. **A draining hemispherical reservoir.** Water is flowing at the rate of $6$ $\text{m}^{3} /\text{min}$ from a reservoir shaped like a hemispherical bowl of radius $13$ m, shown below in profile (side view). Answer the following questions, given that the volume of water in a hemispherical bowl of radius $R$ is $V=(\pi/3) y^{2}(3R-y)$ when the water is $y$ meters deep. ![[1 teaching/smc-fall-2023-math-7/week-8/---files/Pasted image 20231019234029.png]] 1. At what rate is the water level changing when the water is $8$ m deep? 2. What is the radius $r$ of the water's surface when the water is $y$ m deep? 3. At what rate is the radius $r$ changing when the water is $8$ m deep? 6. **The radius of a n inflating balloon.** A spherical balloon is inflated with helium at the rate of $100\pi$ $\text{ft}^{3} /\text{min}$. How fast is the balloon's radius increasing at the moment the radius is $5$ ft? How fast is the surface area increasing? 7. **A balloon and a bicycle.** A balloon is rising vertically above a level straight road at a constant rate of $1$ ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance $s(t)$ between the bicycle and balloon increasing 3 sec later?![[1 teaching/smc-fall-2023-math-7/week-8/---files/Pasted image 20231019234222.png]] 8. **Ships.** Two ships are sailing straight away from a point $O$ along routes that make a $120^{\circ}$ angle. Ship $A$ moves at 14 miles per hour. and ship $B$ moves at 21 miles per hour. How fast are the ships moving apart when $OA=5$ miles and $OB=3$ miles? ////