Quadratic Application Problems
Name:Algebra 2
Date:
Quadratic Application Problems: Geometry Problems
1.Answer the following questions about the rectangle shown in the diagram.- Write a formula for the area of the rectangle in terms of x.
- What value(s) of x make the area of the rectangle equal to 800?
- What is the maximum possible area of the rectangle? What value of x produces that area?
2. / The area of the rectangle drawn at the right is 18 square feet. Find x. /
3.A rectangle has a length of and its width is . 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 and its width is . Find the perimeter of the rectangle.
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 for her pet, with the fence that she has. Answer these questions to determine the best possible design for 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.
6. Many digital cameras take photos that are rectangles with a 4-to-3 ratio. Photos can be enlarged or reduced on a computer, but it’s important to keep the 4-to-3 ratio so that the image isn’t distorted. So the 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 photo?
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?
ANSWERS:
- a.
b.
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.
c. x = 25
6. a.
b. x = 1.5height = 4.5 inchesarea = 27 square inches
c. x = 2.5height = 7.5 incheswidth = 10 inches
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 calculator: What value of x would give the rectangle the largest possible area?
3.Suppose there is a right triangle with side lengths as shown in the picture, where x is an unknown.
a.The Pythagorean Theorem gives a relationship between the three side lengths of the triangle. Fill in this equation with the side lengths in the correct places:
b.Rewrite and simplify the equation by multiplying each of the perfect 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 area of the triangle.