7.1.2How can I model rates of change?

Each situation below involves two different rates that are related to each other. 

1. Ships A and B are attached by a cable of length $z$. As Ship B moves toward point Q, Ship A moves toward point P. Consequently, if $t$ represents time, $\frac { d y } { d t }$ and $\frac { d x } { d t }$ are related rates.​

   1. Describe the speed of Ship A as it approaches point P. Justify your answer.

   2. Are $\frac { d y } { d t }$ and $\frac { d x } { d t }$ each positive, negative, or zero?  

   3. If $\frac { d y } { d t }$ is constant, how does $\frac { d x } { d t }$ change? 

2. A hot air balloon rises vertically from a platform as a rope of length r is slowly released, as shown in the diagram at right. If t represents time, what is the relationship between the rates $\frac { d r } { d t }$ and $\frac { d h } { d t }$? Are they each positive, negative, or zero? If the rate that the rope is released, $\frac { d r } { d t }$, is constant, will the rate that the balloon rises, $\frac { d h } { d t }$, be constant as well? If not, how will $\frac { d h } { d t }$ change?

3. The related rates situations in parts (a) and (b), as well as the boat situation in problem 7-5 each involved one rate that was constant while another rate increased or decreased. Consequently, the related rates were not proportional.  
   
   Examine these diagrams in the three situations closely. What geometric shape do they all have in common? What equations come to mind when examining this shape?

PAINFUL PAINTING, Part One

While painting his house one day, Mr. Cabana had a terrible accident! The bottom of the $20$-foot ladder he was standing on started to slide away from the wall!

1. Draw a diagram of this situation. Then write an equation that relates the sides of that shape in your diagram.  

2. Mr. Cabana wants to know how fast he is falling when the ladder is $10$ feet above the ground. You need to help him solve this problem, but the equation you wrote in part (a) does not involve rates. What Calculus tools can you use to convert the equation in part (a) to a rate equation?

3. Explain why the rate the ladder is sliding down the wall must be related to the rate that the foot of the ladder moving along the ground.

PAINFUL PAINTING, Part Two

Remember from problem 7-14 that Mr. Cabana is standing on top of a $20$-ft ladder when the base starts to slide away from the wall at $1.5$ ft/sec. Mr. Cabana wants to know how fast he will be falling when the ladder is $10$ feet above the ground.

1. Label each side of your diagram from problem 7-14 with a side length and an expression for its rate of change. Substitute any numeric values that are known.

2. Use the strategy you suggested in part (b) of problem 7-14 to convert the equation that relates the sides of the geometric shape into a new equation that relates the rate that each side is changing.  

3. Use your equation from part (b) to answer Mr. Cabana’s question: How fast will he be falling at the moment when the ladder is exactly $10$ feet above the ground?   

4. Explain why the rate you found in part (c) is negative.  

While working with a loop of yarn like you did in problem 7-1, Sophia caused the length $l$ of the rectangle to increase at a rate of $3 \frac { \text{cm} } { \text{second}}$.

Help Sophia calculate the rates below when $l = 15$ cm and $w = 5$ cm. Be sure to include a diagram labeled with appropriate units, and state whether each quantity is increasing, decreasing, or constant.  

1. What is the rate of change of the perimeter of the rectangle?

2. What is $\frac { d w } { d t }$?  

3. Calculate the rate of change of the area of the rectangle.

4. Calculate the rate of change of the length of a diagonal of the rectangle.   

Hustling Harry hastily got the following answer to the indefinite integral below. Verify that he is incorrect, and explain what he did wrong. Homework Help ✎

$\int 2 x \operatorname { cos } ( x ^ { 2 } ) d x = x ^ { 2 } \operatorname { sin } ( \frac { 1 } { 3 } x ^ { 3 } ) + C$

It just so happens that Hustling Harry’s grade in math class at any week $t$ during the semester is calculated by $g(t)=46\cos(\frac{t}{10})+10\sin(\frac{t}{2})+40$. At what point during the $18$-week semester is Harry’s grade equal to his average grade for the semester? Homework Help ✎

A point travels along the $x$-axis so that at time $t$ its position is given by $s(t) = t^3 - 5t^2 + 4t$, where $0 ≤ t ≤ 5$. Homework Help ✎

1. ​What does $s(t)$ represent?

2. Where is the particle located when $t = 5$?

3. What is the velocity of the particle when its acceleration is zero?

Patrick, who is $6$ feet tall, is walking way from a $10$-foot tall lamppost. If he currently casts a shadow that is $12.5$ ft long, how far from the lamppost is he? Homework Help ✎

Solve for $y$ if $\frac { d y } { d x }= 3x^{−1/4}− \frac { 7 } { 2 }\sqrt { x }$. Homework Help ✎

Explain how the first derivative can be used to determine a local minimum or a local maximum. Be sure to distinguish between the two. (Note: What if the first derivative is undefined?) Homework Help ✎