4 Calc Problems

MATH 160 Name:

Written Assignment CSU ID: Equations from Physical Descriptions Please print clearly

Assigned: Tuesday, October 18th Section 801 (Benoit) Due: Monday, October 24th Fall, 2016

Write your solutions in the spaces provided. If you need more paper, attach additional sheets, but make a note on the problem to let the grader know to look for more work on extra sheets.

Scan and email your solutions to your instructor (smartphone photos are fine if all work is readable).

This assignment focuses on extracting mathematical equations or inequalities from a verbal descrip- tion - a key skill to have when doing related-rates or optimization problems.

Geometric formulas you will be expected to know:

• The area of a rectangle, triangle, circle (and more complex shapes that can be divided up into these basic shapes).

• The volume of a rectangular box, right cylinder (not necessarily circular), cone, and sphere.

• The surface area of a rectangular box, right circular cylinder, and sphere.

• The distance between two points given their (x, y) coordinates.

• The perimeter of a circle, and be able to calculate perimeter of shapes made up of line segments and semicircles or quarter circles.

Instructions:

For each of the descriptions on the following pages, write down a list of variables, and describe what each variable represents, and then write down an equation or inequality that expresses the relationship described. You do not need to try to solve the resulting equation.

Example:

Statement: The volume of a sphere is half of the product of its surface area and its height above the table. Solution:

r = the radius of the sphere

V = 4

3 ⇡r

3 = the volume of the sphere

A = 4⇡r2 = the surface are of the sphere

h = the height of the sphere above the table

V = 1

2 Ah or

4

3 ⇡r

3 = 1

2 4⇡r2h

1. A conical tank that is 10 feet in height is being filled with water. The bottom of the tank is a sharp point, and the top is a circle 10 feet across. Find the volume of water in the tank as it fills as a function of its depth.

2. As pancake batter is poured onto a griddle, it forms a circle 1/4 of an inch thick. Find the circumference of the pancake as a function of the amount (volume) of batter that has been poured on.

3. A 12 foot tall pole standing vertically casts a shadow 8 feet long on the ground. As the pole falls over, directly away from the sun, find the length of the shadow as a function of the pole’s angle away from vertical.

4. Two ships are sailing in clear water. Ship 1 starts at coordinates (10, 5) and moves left (in the �x direction) at 1 unit per second. Ship 2 starts at the origin, and moves up (in the +y direction) at 2 units per second. Find the distance between the ships as a function of time.