Setting Up Functions

Sol 1

1. The perimeter is given by .

2. , so .

3. Substituting back gives .

Sol 2

1. The area is given by .

2. The fencing satisfies , so .

3. Substituting back gives .

Sol 3

1. The area is given by .

2. A diagonal of the rectangle will be a diameter of the circle, so its length is 10. Then by the Pythagorean Theorem, so and since .

3. Substituting back gives .

Sol 4

1. The area of the rectangle is given by .

2. Since the upper right vertex is on the given parabola, we have that .

3. Substituting back gives .

Sol 5

1. The area of the triangle is given by .

2. The slope of the hypotenuse is given by

so solving for gives

so

3. Substituting back gives

.

Sol 6

1. Since the two semicircular regions can be combined to give a circular region, the area of the field is given by .

2. Since the perimeter is 400 meters, so and .

3. Substituting back gives .

Sol 7

1. The area of the page is given by .

2 We have that and , and that the area of the printed material is given by , so .

3. Substituting back gives .

Sol 8

1. Since the top and bottom each have area given by and the other 4 sides each have area given by , the total surface area is given by .

2. Since the volume is 80 cubic inches, and therefore .

3. Substituting back gives

.

Sol 9

1. The cost of the top and bottom is given by , and the cost of the other 4 sides is given by , so the total cost is expressed by .

2. Since the volume is 60 cubic inches, and therefore .

3. Substituting back gives .

Sol 10

1. Since the top and the bottom each have area , the total surface area is given by .

2. The volume of the cylinder is the area of the base multiplied by the height, so

and .

3. Substituting back gives .

Sol 11

1. The cost of the top and bottom is given by , and the cost of the side is given by , so the total cost is given by .

2. The volume of the cylinder is the area of the base multiplied by the height, so

and .

3. Substituting back gives .

Sol 12

1. We know that for the cylinder.

2. The total surface area is square inches, so and therefore

. Solving for gives and .

3. Substituting back gives

.

Sol 13

1. We know that , where

2. by the Pythagorean Theorem, so .

Using similar triangles, , so .

3. Substituting back gives .

Sol 14

1. We have that , where is the time he walks off the road and is the time he walks along the road.

2. Using the formula , we get that ; so

and .

3. Substituting back gives .

Lawrence Marx 2013-09-23