Applications

Example 1

An object given an initial velocity of 48 ft/sec has its height h given by [Maple Math] . Find the time when the object reaches its maximum height and find the maximum height.

Solution .

Since a = -16, the parabola does open down and has a maximum point. The t value at the vertex and the time at which it reaches its maximum height is [Maple Math] or 1.5. Putting t = 1.5 into h gives a maximum height of 36 ft.

[Maple Plot]

Example 2

The Wind Up Company sells a dancing doll for $50. If the cost to produce x dolls is [Maple Math] , find the profit function and the number of dancing dolls that maximizes the profit.

Solution .

Since profit is revenue minus cost, we must first find the revenue function. Assuming each doll is sold for $50, the revenue is 50x. Profit then equals [Maple Math] or [Maple Math] . Since a<0, there is a maximum. Using [Maple Math] the x value at the vertex is 4000. The maximum profit is P(4000) or $35000.

Example 3

A person wants to enclose the largest rectangular area using 300 feet of fencing on three sides. The fourth side will not be fenced in since it is along side a body of water. Find the dimensions of such a rectangle that encloses the largest area.

Solution .

[Maple OLE 2.0 Object]

We need to maximize A = xy. Since 300 = x + 2y, we can write x = 300 - 2y. We can then write A = (300 -2y)y or [Maple Math] . Since a = -2 is less than zero, the function opens down and hence has a maximum at the vertex. To start using completing the square we write [Maple Math] . Since the square of half of -150 is 5625, we write

[Maple Math] . ( Note that the 11250 balances the negative two times 5625 .)

Factoring the trinomial, we get [Maple Math] . The area is a maximum of 11250 square feet, when y = 75 feet. Since x = 300 -2y, x = 150 feet. The required dimensions are 150 feet by 75 feet.

Example 4

A quadratic function has vertex at ( 2, -5) and goes through the point (4, 7). Find the equation of the function.

Solution .

The vertex form of a quadratic function is [Maple Math] . Since V= (2, -5) gives us the values of h and k we have [Maple Math] . Because the curve goes through (4, 7), we know that f(4) = 7. Substituting we have [Maple Math] or [Maple Math] . Solving this give us a = 3. Consequently the function is [Maple Math]

[Maple Plot]