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Three sides of a trapezoid are each 10 cm long.
How long is the 4th side, when the area of the trapezoid has the greatest value?
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It is clear, that under given conditions, the trapezoid is isosceles with the shorter base of 10 cm long.
See my sketch below. The upper base is 10 cm; the lower base is 10+2x cm long.
10
+------------------+
/| \
/ | \
10 / | \ 10
/ | \
/ | \
+------------------------------+
<- x ->
The mid-line is = 10+x cm long.
The height is .
The area of the trapezoid is A(x) = .
To find the maximum of A(x), take the derivative
= -
= -
Equate it to zero
- = 0.
Simplify and find x
=
= (10+x)*x
=
= 0
= 0
(x-5)*(x+10) = 0
Of two roots, x= 5 and x= -10, take the positive one, which gives the maximum to the trapezoid's area.
ANSWER. The area of trapezoid is maximum when the 4-th side is 10+2*5 = 20 cm long.
Solved.