Math 80 – Modeling Using Quadratic / Polynomial Functions
1) Physics / Projectile Motion. A baseball is hit by a batter. The height (in feet) h(t) of the ball after t
seconds is given by ℎ 𝑡 = −16𝑡 ! + 125𝑡 + 4 . Use a completing the square or quadratic formula
approach to find the following (round to the nearest tenth).
a) What is the height of the ball after 2 seconds?
b) When is the ball at a height of exactly 200 feet?
c) When does the ball reach the ground?
2) Economics / Revenue. A group charters a flight that normally costs $800 per person. A group
discount reduces the fare by $10 for each ticket sold; the more tickets sold, the lower the per-person
fare. There are 60 seats on the plane.
a) Give the equation for the total revenue earned by the airline R(n), where n is the number of
tickets sold.
b) Find the total revenue for the airline should the group be of size n = 35.
c) What size group would maximize the airline’s revenue?
3) Geometry / Architecture. The length of a rectangular enclosure is 6 meters more than the width. If
both the length and width where doubled, the area would be 108 square meters. Find the dimensions
of the original rectangle.
4) Geometry / Architecture. A flat wooden frame of uniform width, with outside dimensions of 8 feet
by 10 feet, surrounds a rectangular painting at the Ghetty Center. If 15 square feet of the painting is
visible, how wide is the material from which the frame is made?
5) Projectile Motion. Let f(t) be the height (in feet) of a stone at t seconds after launched into the air.
A reasonable equation of f is 𝑓 𝑡 = −16𝑡 ! + 80𝑡 + 3. At what time is the ball at its maximum height?
What is that height?
6) Physics / Projectile Motion. A person on the edge of a cliff throws a stone so that it hits the ground
near the base of the cliff. The stone’s height (in feet above the base of the cliff) h(t) after t seconds is
given by ℎ 𝑡 = −16𝑡 ! + 30𝑡 + 200.
a) Find the maximum height of the stone. How long does it take for the stone to reach its
maximum height?
b) How tall is the cliff?
c) Is the stone being thrown upward or downward? Explain.
d) How long does it take for the stone to fall to ground level (that is, at the base of the cliff)?
7) Geometry / Architecture. An architect plans to use 1200 feet of fence and the side of a barn to form
a rectangular enclosure for farm animals. (That is, a 3 sided rectangular enclosure – the 4th side uses
the barn itself). What dimensions of the rectangle would maximize the area?
8) Economics / Revenue. A company charters a party boat that normally costs $60 per person. A
group discount reduces the fare by $0.50 for each ticket sold; the more tickets sold, the lower is the
per-person fare. The maximum capacity of the boat is 80 people. What size of group would maximize
the boat company’s revenue?