Quadratic functions are useful as mathematical models. We will use them to model objects experiencing the force of gravity, or for modeling income when demand is a linear function of the price.
A flower pot falls from the ledge of a balcony on a high-rise building. An object experiencing the force of gravity can be modeled by the equation \(h(t)=-16t^2+vt + c\text{,}\) where \(t\) is the time in seconds, \(h(t)\) is the height in feet, \(c\) is the initial height of the object, and \(v\) is the initial velocity of the object. Note: This model applies to any object experiencing the force of gravity, where the measurement is in English units (feet). There is another version using metric units (meters). This model also assumes air resistance is negligible, so it doesn’t work well for things like feathers and parachutes where air resistance is strong.
Esperanza owns an independent motel that has 50 rooms. She finds that if she charges $40 per night, all the rooms will be rented. Thereafter, for every $4 she raises the room rate, 2 fewer rooms will be rented out.
Make a table showing the price of the room, the number of rooms rented, and the total income for the hotel for prices of $40 to $60 per night, in $4 increments.
Suppose Esperanza will be satisfied if the motel’s income for a night is at least $2300. What are the possible prices she can charge to earn this income?
Adam is raising pigs on his farm. He needs to build a rectangular pen for his pigs, and he wants to give them as much area as possible. However, he only has 180 feet of fencing.
Ernesto wants to make an open-top cardboard box to store some items on his desk. His original piece of cardboard is 9 inches by 12 inches. He needs to cut out squares from each corner of the cardboard and fold up the resulting sides to make a box, as in Figure 1.5.6. He wants to create the largest volume possible for his box.
