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cgarnett5408
28.11.2019 •
Mathematics
Arectangle is inscribed in an equilateral triangle so that one side of the rectangle lies on the base of the triangle. find the maximum area the rectangle can have when the triangle has side length 14 inches.
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Ответ:
A(max) = 42.43 in²
Dimensions:
a = 7 in
b = 6,06 in
Step-by-step explanation: See annex
Equilateral triangle side L = 14 in, internal angles all equal to 60°
Let A area of rectangle A = a*b
side b tan∠60° = √3 tan∠60° = b/x b = √3 * x
side a a = L - 2x a = 14 - 2x
A(x) = a*b A(x) = ( 14 - 2x ) * √3 * x
A(x) = 14*√3*x - 2√3 * x²
Taking derivatives both sides of the equation
A´(x) = 14√3 - 4√3*x
A´(x) = 0 ⇒ 14√3 - 4√3*x = 0 ⇒ 14 - 4x = 0 x = 14/4
x = 3,5 in
Then
a = 14 - 2x a = 14 - 7 a = 7 in
b = √3*3,5 b = *√3 *3,5 b = 6,06 in
A(max) = 7 *6,06
A(max) = 42.43 in²
Ответ:
Yes because of the way the it is.
Step-by-step explanation: