1
Algebra II – Chapter 4 Day #3
Topic: Modeling with Quadratic Functions
Today, we will begin looking at word problems that involve many of the concepts we have
tackled with regards to quadratic functions.
1. Among all pairs of numbers (x, y) such that 3x + y = 11 find the pair for which the sum of the
squares is a minimum.
2. Find the pairs of numbers (x, y) whose product is maximum if three times the first number
plus the second number is 60.
3. The total revenue of Tran’s Machinery Rental is given as the function: R(x) = 300𝑥 − 0.4𝑥 2
where x is the number of units rented. What number of units rented produces the
maximum revenue? What is the maximum revenue?
4. Among all rectangles that have a perimeter of 60, find the dimensions of the one whose
area is the largest.
5. A gardener is putting a wire fence along the edge of his garden to keep animals from eating
his plants. If he has 20 meters of fence, what is the largest rectangular area he can enclose?
6. The back of a farmer’s barn is going to serve as one side of the fence that will enclose a
rectangular area. He will need fencing for the other 3 sides of the lot. If he has 500 feet of
fencing, what is the maximum possible area of the lot?
7. STEM: Engineering:
The New River Gorge Bridge in West Virginia is the world’s largest steel single arch bridge.
You can model the arch with the function:
𝒚 = −𝟎. 𝟎𝟎𝟎𝟒𝟗𝟖𝒙𝟐 + 𝟎. 𝟖𝟒𝟕𝒙,
where x and y are in feet. How high above the river is the arch? How long is the section of
bridge above the arch?
2
8. STEM: Landscaping: A town is planning a playground. It wants to fence in a rectangular
space using an existing wall. What is the greatest area it can fence in using 100 feet of
donated fencing?
9. STEM: Economics: A model for a company’s revenue from selling a software package is:
R = −2.5𝑝2 + 500𝑝, where p is the price in dollars of the software.
a. What price will maximize revenue?
b. Find the maximum revenue?
10. STEM: Physics: The height of a projectile fired straight up in the air with an initial velocity of 64 ft/s
is h = 𝟔𝟒𝒕 − 𝟏𝟔𝒕𝟐 , where h is the height in feet and t is time in seconds. The table represents the
data for another projectile.
Time(t)
Height (h)
0.5
20
1
1.5
2
32
36
32
a. Which projectile goes higher? How much higher?
b. At what times ‘t’ will each projectile be at a height of 16 feet?
3
HOMEWORK – Chapter 4 Day #3
1. Among all the pairs of numbers (x, y) such that 2x +y = 20, find the pair for which the
sum of the squares is minimum.
2. The back of George’s property is a creek. George would like to enclose a rectangular
area, using the creek as one side and fencing for the other three sides, to create a
pasture for his two horses. If he has 300 feet of material, what is the maximum possible
area of the pasture?
3. Find a pair of numbers whose product is a maximum if two times the first number plus
the second number is 48.
4. The total revenue for Thompson’s Studio Apartments is given as the function
R(x) = 𝟏𝟎𝟎𝒙 − 𝟎. 𝟏𝒙𝟐 ,
Where x is the number of rooms rented. What number of rooms rented produces the
maximum revenue?
5. The total Revenue of Tran’s Machinery Rental is given as the function
R(x) = 𝟑𝟎𝟎𝒙 − 𝟎. 𝟒𝒙𝟐 ,
Where x is the number of units rented. What number of units rented produces the
maximum revenue?
6. The total cost of producing a type of small car is given by
C(x) = 𝟗𝟎𝟎𝟎 − 𝟏𝟑𝟓𝒙 + 𝟎. 𝟎𝟒𝟓𝒙𝟐,
Where x is the number of cars produced. How many cars should be produced to incur
minimum cost?
7. STEM: Engineering: The Gateway Arch in St. Louis was built in 1965. It is the tallest
monument in the United States. The arch can be modeled with the function:
y = 0.00635x2 + 4x, where x and y are in feet.
a. How high above the ground is the tallest point of the arch?
b. How far apart are the legs of the arch at their bases?
4
8. STEM: Sports: The height of a batted ball is modeled by the function:
h = 0.01x2 + 1.22x + 3, where x is the horizontal distance in feet from the point of
impact with the bat, and h is the height of the ball in feet.
a. What is the maximum height that the ball will reach?
b. At what distance from the batter will the ball hit the ground?
9. STEM: Engineering: The Zhaozhou Bridge in China is the oldest known arch bridge,
dating to A.D. 605. You can model the support arch with the function:
f(x) = −0.001075𝑥 2 + 0.131148𝑥, where x and y are measured in feet. How high is the
arch above its supports?
10. STEM: Forestry: A local nursery sells a large number of ornamental trees every year.
The owners have determined the cost per tree C for buying and caring for each tree
before it is sold is C = 0.001n2  0.3n + 50. In this function, C is the cost per tree in
dollars and n is the number of trees in stock.
a. How many trees will minimize the cost per tree?
b. What will the minimum cost per tree be?