Applications of Quadratic Functions

  1. A manufacturer of fax machines find that the cost (in dollars) generated by manufacturing x units per week is given by the function How many units should be manufactured to minimize the cost? Show this both algebraically and graphically.

-- The minimum of this quadratic function is the vertex. So the number of units that should be manufactured are x=-b/(2a)=39/.3=130. To show this graphically, you would just have to find the correct window on your calculator and use the min/max option.

  1. The main cable of a suspension bridge has the shape of a parabola. The cables are strung from the tops of two towers, 200 feet apart, each 50 feet high. The cable is 5 feet above the roadway at the point that is directly between the towers (see the figure). A. Determine an equation for the parabola. B. How long is the vertical support cable that is 40 feet from one of the towers?

--The vertex of the parabola is at (100,5) and another point is (0,50). Using vertex form you get h(x)=(45/10000)(x-100)^2 + 5. The length of a vertical support cable is the same as the height of the cable from the ground. So when x=40, the height is (45/10000)*3600+5

  1. The arch of a doorway is in the shape of a parabola. The bottom of the arch is 4 feet above the floor, and the top of the arch is 8 feet above the floor. The doorway is 4 feet wide (see figure). A. Determine the equation for the parabola. B. Is it possible to pass a box with dimensions feet through the doorway?

--Your equation depends on where you put the origin. Here is one possible solution: vertex is at (2,8) and another point is (0,4). The vertex form becomes: h(x)=-(x-2)^2+8

The 10 foot side of the box will go through lengthwise. Now we need to determine whether 3 feet wide and 6 feet high will fit through. If we feed it through the middle, the side of the box is .5 foot from the side of the door. The height of the door at this point is h(.5)=5.75 feet, so not quite high enough. The box won’t fit through.

  1. What is the greatest area of a pasture in the shape of a rectangle that can be fenced with 6000 feet of fencing?

--A rectangle with length l and width w has area of A=wl. The perimeter is 6000 which, in terms of l and w is 6000=2l+2w. If you solve for w, you get w=3000-l. Substitute this into your area function: A=(3000-l)*l. This is quadratic, with is maximum at l=3000/2=1500 ft. So when l=1500 the area is maximized. If l=1500, w=3000-1500=1500. And the area is A=1500*1500=2250000 square feet.

  1. A rain gutter is constructed by folding the edges of a sheet of metal 12 inches wide so that the cross section of the gutter is a rectangle (see figure). The capacity of the gutter is the product of the length of the gutter times the area of the cross section. How much edge should be folded up on each side to maximize the capacity?

--Think of the picture I drew on the board. I’ll label the length of the bottom b, and the length of an edge e. The length of the gutter is some constant, let’s call it l.

We want to maximize capacity, so we need to come up with an expression for C in terms of our variables: the area of a cross section is be, and so C=be*l. We also know 12=b+2e. So, b=12-2e. Substitute this into your Capacity equation: C=(12-2e)e*l. This is a quadratic function if we remember that l is just a constant. It’s maximum is at e=(-12l)/(2*-2l)=3. So when the length of an edge is 3, the capacity is maximized.

  1. Suppose that Determine a value for k such that the graph of f has: A. No x-intercepts. B. Exactly one x-intercept. C. Two x-intercepts.

--The intercepts of f(x) can be found by using the quadratic formula: x=(k+-sqrt(k*k-64))/2. If what’s under the square root is less than zero, there are no solutions. If it’s equal to zero, there is one intercept and if it’s greater than zero, there are two. For k=8 or k=-8 there is one intercept, for -8<k<8 there are no solutions and for k<-8 or k>8 there are two solutions.

  1. Suppose that Determine a value for c such that the graph of f has: A. No x-intercepts. B. Exactly one x-intercept. C. Two x-intercepts.

--Same procedure as 7.