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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 3: Q. 18 (page 166)

Architecture A special window has the shape of a rectangle surmounted by an equilateral triangle. See the figure. If the perimeter of the window is $16$ feet, what dimensions will admit the most light?
[Hint: Area of an equilateral triangle $(\frac{\sqrt{3}}{4}) x^{2},$ where $x$ is the length of a side of the triangle.]

Short Answer

Expert verified

The dimension will admit the most at $14.95 f t^{2} .$

Step by step solution

01

Step 1. Given information

It is given that a special window has the shape of a rectangle surrounded by an equilateral triangle if the perimeter of the window is $16 f t .$ We need to determine the dimension that will admit the most.

02

Step 2. Solving

The perimeter of the window is calculated by,

$P = 3 x + 2 a .$

Put $P = 16$ in $P = 3 x + 2 a .$

$16 = 3 x + 2 a .$

$2 a = 16 - 3 x .$

$a = 8 - \frac{3}{2 x} .$

The area of the window $(A)$ is determined by the sum of the area of an equilateral triangle $(A_{t})$ and the area of a rectangle.

$(A_{t}) = \frac{\sqrt{3}}{4 x^{2}} .$

$(A_{r}) = a 路 x .$

$A = (A_{t}) + (A_{r}) .$

localid="1647342292315" $\frac{= \sqrt{3}}{4 x^{2} + a x .}$

$= \frac{\sqrt{3}}{4 x^{2} + (8 - \frac{3}{2 x}) x .}$

$= \frac{\sqrt{3}}{4 x^{2} + 8 x - \frac{3}{2 x^{2}} .}$

$A = - 1.07 x^{2} + 8 x .$

The vertex of the function is determined by,

$x = - \frac{8}{2 路 (- 107)} .$

$= 3.74 f t .$

For the value of $x = 3.74 f t,$ the maximum window area is,

$A = - 1.07 {(3.74)}^{2} + 8 (37.74) .$ $A = - 1.07 {(3.74)}^{2} + 8 (37.74) .$

$= 14.95 f t^{2} .$

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