2.7 Mathematical Models
Some will win, some will lose, some are born to sing the blues.
Oh the movie never ends, it goes on and on and on and on.
-Journey
Modeling Volume
An open box with a square base is to be cut from a square piece of
cardboard with side equal to 18 inches by cutting out a square
from each corner and turning up the sides.
1) Write the volume V of the box as a function of the length x of
the square cut from the box.
Modeling Area
A farmer has 3000 feet of fencing to enclose a rectangular field.
One side lies along a river, so only three sides need fencing.
1) Express the area A of the field enclosed by the fencing as a
function of l, the length of the side parallel to the river.
Modeling Distance
Let P = (x,y) be a point on the graph of 𝑦 = 𝑥 2 − 1
1) Express the distance d from P to the origin as a function of x.
2. What is d if x = 0?
3. What is d if x = 1?
Modeling Area
A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be
the point in Quadrant I that is a vertex of the rectangle and is on
the circle.
1) Express the area A of the rectangle as a function of x.
Modeling Area
A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be
the point in Quadrant I that is a vertex of the rectangle and is on
the circle.
2) Express the perimeter P of the rectangle as a function of x.
4  x2
2x

Modeling Area
A rectangle has one corner on the graph 𝑦 = 25 − 𝑥 2 , another at
the origin, and a third on the positive y-axis and the fourth on the
positive x-axis.
A) Express the area A of the rectangle as a function of x.
B) What is the domain of A?
C) Graph A=A(x)
Economics: Demand Equations
In economics, revenue, R, in dollars, is defined as the amount of
money received from the sale of a product and is equal to the
Unit selling price p, in dollars, of the product times the number
X of units actually sold.
R = xp
Economics: R = xp
The price p in dollars and the quantity x sold of a certain product
obey the demand equation.
1. Express the revenue R as a function of x.
2. What is the revenue if 100 units are sold?
3. What quantity x maximizes revenue? What is the maximum
revenue?
4. What price should the company charge to maximize revenue?
2.7 Mathematical Models
Homework: p.134
#1, 3, 7a, 9abc, 11a,
15ab, 21 & 22
Some will win, some will lose, some are born to sing the blues.
Oh the movie never ends, it goes on and on and on and on.
-Journey