7.1.4Why are good diagrams so important?

Related Rates Applications: Choosing the Best Formula

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Solving Related Rate Problems

The basic process for solving any related rates problem is:

Sketch a diagram and use geometry to identify a relationship between the changing quantities. For example, is there an appropriate area or volume formula that you can apply? Is there a right triangle for which the Pythagorean Theorem can be applied? Are there similar triangles?
Use implicit differentiation to write a rate relationship between the quantities. Do not forget to differentiate with respect to time (if time is important)!
Substitute the given values of the quantities and rates to solve for the desired rate. Be sure to make any decreasing rates negative.

7-34.

Happy the Clown has balloons that are perfect spheres. Happy fills the balloons with helium at a rate of $20\text{ in}^3$ per second. The recommended volume for each balloon is $500 \text{ in}^3$ . Happy wants to know how fast the diameter of a balloon is changing when the volume is at $500 \text{ in}^3$ . Happy is now sad because he knows his answer shown below does not make sense. Find and fix his error so that Happy will be happy again.

$\left. \begin{array}{l}{ V = \frac { 4 } { 3 } \pi r ^ { 3 } }\\{ 500 = \frac { 4 } { 3 } \pi r ^ { 3 } }\\{ 0 = 4 \pi r ^ { 2 } \frac { d r } { d t } }\\{ 0 = \frac { d r } { d t } }\end{array} \right.$

7-35.

LATE TO CLASS

Sophorn and Jonathan are late to class again, but this time the principal has caught them! Sophorn, who is at the soda machine buying a soda for her teacher, hustles toward the principal at $4$ ft/sec, waving her hall pass. At that same instant, Jonathan, who is standing in front of the principal’s office without a pass, escapes from the principal, running towards his classroom at a speed of $6$ ft/sec.

Make a prediction: The distance between Sophorn and the principal’s office is decreasing while the distance between Jonathan and the principal’s office is increasing. Describe the distance between Sophorn and Jonathan. Is it increasing, decreasing, or does it stay constant? Explain your thinking.
Make another prediction: Both Sophorn and Jonathan run at constant rates. Does this mean that the distance between them also changes at a constant rate? Explain your thinking.
Evaluate and compare the distances between Sophorn and Jonathan at $t = 0, 1$ , and $3$ seconds. Then evaluate and compare the rates that the distance between Sophorn and Jonathan changes at $t = 0, 1$ , and $3$ seconds. Note that at $t = 0$ , Jonathan is at the same position as the principal and the soda machine, where Sophorn starts, is $15$ feet from the principal.

7-36.

A punch glass is in the shape of a hemisphere with radius of $5$ cm. If punch is being poured into the glass so that the change in height of the punch is $1.5$ cm/sec, at what rate is the exposed area of the punch changing when the height of the punch is $2$ cm?

7-37.

As clumsy Kenny pours his cold coffee on the floor, he notices that the growing puddle is circular. Let $A$ be the area of a circle with radius $r$ at time $t$ . Write an equation that relates $\frac { d A } { d t }$ to $\frac { d r } { d t }$ . Homework Help ✎

7-38.

Let $S$ be the surface area of an inflatable cube with sides of length $x$ at time $t$ . Write an equation that relates $\frac { d S } { d t }$ to $\frac { d x } { d t }$ . Homework Help ✎

7-39.

Without a calculator, determine the minimum value of $f(x) = 2x^3 - 15x^2 + 24x + 19$ for $0 ≤ x ≤ 5$ . Use the second derivative to justify that your value is a minimum. Homework Help ✎

Compute without a calculator

7-40.

Two roads intersect at right angles. Elizabeth, who is driving east, leaves the intersection traveling at a speed of $40$ mph. Two hours later, Nairi leaves the same intersection at a speed of $30$ mph heading north. How fast is the distance between the cars changing five hours after Elizabeth leaves? Homework Help ✎

7-41.

Write the equation of the line(s) tangent to $y = x^2 + 2x + 4$ that pass through the origin. 7-41 HW eTool Homework Help ✎

7-42.

Kimberly and Varag are in a bicycle race. Homework Help ✎

If Kimberly’s velocity in miles per hour during the race is $v(t) = 2.3^t\sin( \frac { t } { 2 } )$ , calculate her average velocity during $0 ≤ t ≤ 6$ . Describe your method.
If Varag’s distance from the starting line during the race is $s(t) = 20t + 2\sin(t)$ , calculate his average velocity during $0 ≤ t ≤ 6$ . Describe your method.
When is each bicyclist traveling at his/her average velocity?

7-43.

Consider the equation $xy^2 - x^3y = 12$ . Homework Help ✎

Show that $\frac { d y } { d x }=\frac { 3 x ^ { 2 } y - y ^ { 2 } } { 2 x y - x ^ { 3 } }$ .
Determine the $x$ -coordinate of each point on the curve where the tangent line is vertical.

Calculus Third Edition

Solving Related Rate Problems