Quadratic Application Problems

Name: Algebra 2 Date: Quadratic Application Problems: Geometry Problems

50–x 1. Answer the following questions about the rectangle shown in the diagram. 2x

a. Write a formula for the area of the rectangle in terms of x.

b. What value(s) of x make the area of the rectangle equal to 800?

c. What is the maximum possible area of the rectangle? What value of x produces that area? Name: Algebra 2 Date: 2. The area of the rectangle drawn at the right is 18 square feet. Find x. x+3

3. A rectangle has a length of x + 9 and its width is x + 7. Find the dimensions of the rectangle if its area is 143 square meters.

4. The area of a rectangle is 91 square centimeters. Its length is x + 2 and its width is x - 4. Find the perimeter of the rectangle.

Name: Algebra 2 Date:

5. Mrs. Meriam wants to make a pet enclosure next to her house. She will use her house as one side of a rectangle, and build a fence forming the other three sides, as shown in the diagram. Let x stand for two of the side lengths, as shown. In total, Mrs. Meriam has 100 feet of fence to use. She wants to build the largest possible enclosure f or her pet, with the fence that she has. Answer these questions to determine the best possible design f or her to use. a. What is the length for the unlabeled side of the rectangle?

b. Write a function formula for f(x), the area of the pet enclosure.

c. Find the answer to this question algebraically, then check it using graphing on your calculator: Find the value of x that creates the largest possible area.

d. Write a set of instructions for Mrs. Meriam about how she should build the fence. Name: Algebra 2 Date: 6. Many digital cameras take photos that are rectangles with a 4-to-3 ratio. Photos can be enlarged or re duced on a computer, but it’s important to keep the 4-to-3 ratio so that the image isn’t distorted. So t he dimensions of a photo would be like this:

where x could be any number. a. Write a formula for f(x), the area of the photo as a function of x.

b. If the photo is 6 inches wide, what is the value of x, and what are the height and the area of the p hoto?

c. Answer this question by writing an equation and solving it algebraically: Suppose the photo’s area is 75 square inches. What is the value of x, and what are the height and the width of the photo? Name: Algebra 2 Date:

ANSWERS: 1. a. A(x)= 2x(50 - x) b. x = 40 or x = 10 c. Area is 1250 for x = 25 2. x = 6 3. The rectangle is 13 by 11. (x = 4). 4. Perimeter = 40. 5. a. 100 – 2x

b. f (x) = x(100 - 2x)

c. x = 25

6. a. f (x) = (3x)(4x) =12x 2

b. x = 1.5 height = 4.5 inches area = 27 square inches

c. x = 2.5 height = 7.5 inches width = 10 inches Name: Algebra 2 Date: 2. Suppose that a rectangle has a perimeter of 20 inches. Let x stand for the rectangle’s height. a. Given the above information, explain why the rectangle’s width must be (10 – x).

b. Sketch a picture of the rectangle labeled with its dimensions.

c. Let f(x) stand for the rectangle’s area. Write a function formula for f(x). Then rewrite f(x) in the form ax2 + bx + c.

d. Answer this question by writing an equation and solving it algebraically: What value of x would make the rectangle’s area equal 21 square inches?

e. Find the answer to this question by finding the maximum of a function, without using your calcul ator: What value of x would give the rectangle the largest possible area? Name: Algebra 2 Date: 3. Suppose there is a right triangle with side lengths as shown in the p icture, where x is an unknown. x+6 a. The Pythagorean Theorem gives a relationship between the thr x-2 ee side lengths of the triangle. Fill in this equation with the side lengths in the correct places: ( )2 + ( )2 = ( )2 x+5 b. Rewrite and simplify the equation by multiplying each of the p erfect squares, then combining like terms and simplifying as much as possible.

c. Algebraically solve the above equation for x.

d. You should have gotten two x values when you solved the equation, but only one of them makes sense in terms of the triangle. Tell why the other x value can be eliminated as an answer.

e. Using the x value that is a valid solution, find the three side lengths of the triangle, then find the a rea of the triangle.