You can
put this solution on YOUR website!1st one: A lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg??
.
Let x = length of longer leg
then
x+60 = hypotenuse
.
Applying pythagorean's theorem:
180^2 + x^2 = (x+60)^2
180^2 + x^2 = (x+60)^2
32400 + x^2 = x^2 + 120x + 3600
32400 = 120x + 3600
28800 = 120x
240 meters = x (longer leg)
.
2nd one: A rectangular piece of cardboard measuring 28 inches by 29 inches is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each such square. For what value of x will the volume be a maximum? If necessary, round to 2 decimal places.
.
Volume = x(28-2x)(29-2x)
Volume = x(812-56x-58x+4x^2)
Volume = x(812-114x+4x^2)
Volume = 812x-114x^2+4x^3
Volume = 4x^3-114x^2+812x
.
Taking the derivative:
4x^3-114x^2+812x
12x^2-228x+812 = 0
6x^2-114x+406 = 0
Solving the above with the quadratic equation we get:
x = {14.25, 4.75}
.
Solution: 4.75 inches
.
Details of quadratic equation:
| Solved by pluggable solver: SOLVE quadratic equation with variable |
Quadratic equation  (in our case  ) has the following solutons:
For these solutions to exist, the discriminant  should not be a negative number.
First, we need to compute the discriminant :  .
Discriminant d=3252 is greater than zero. That means that there are two solutions:  .
Quadratic expression  can be factored:
Again, the answer is: 14.2521924764611, 4.74780752353892.
Here's your graph:
 |
