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If three sides of a trapezoid are 6 in. long, how long must the fourth side be if the area is maximum.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
Since the altitude of the trapezoid is not given in the problem, then label further the above figure as follows
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| Photo by Math Principles in Everyday Life |
Apply Pythagorean Theorem at the two right triangles of a trapezoid in order to get the value of h, we have
We know that the area of a trapezoid is
Substitute the values of h, b1, and b2 to the above equation, we have
Take the derivative on both sides of the equation with respect to x, we have
Set dA/dx = 0 because we want to maximize the area of a trapezoid
Equate each factor to zero and solve for the value of x:
If
Since the value of x is negative, then it is not accepted as a part of a length of a base of a trapezoid.
If
Since the value of x is positive, then it is accepted as a part of a length of a base of a trapezoid.
Therefore, the length of the fourth side of a trapezoid is
