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
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 .