Q. 497 A daycare facility is enclosing ... [FREE SOLUTION]

91影视

Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q. 497 (page 986)

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using $180$ feet of fencing on three sides of the yard. The quadratic equation $A = - 2 x^{2} + 180 x$ gives the area $A$ , of the yard for the length, $x,$ of the building that the border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximize area.

Short Answer

Expert verified

The length of the building that should border the yard to maximize the area is $45$ ft. The maximum area is role="math" localid="1645262091874" $4050 {ft}^{2}$ .

Step by step solution

Step 1. Given information.

The given equation is

$A = - 2 x^{2} + 180 x$

Step 2. Find the maximum area.

We have:

$A (x) = - 2 x^{2} + 180 x$

$The 鈥� vertex 鈥� of 鈥� an 鈥� up - down 鈥� facing 鈥� parabola 鈥� of 鈥� the 鈥� form 鈥� y = a x^{2} + b x + c 鈥� is 鈥� x_{v} = - \frac{b}{2 a}$

$The 鈥� parabola 鈥� params 鈥� are : a = - 2, 鈥� b = 180, 鈥� c = 0 x_{v} = - \frac{b}{2 a} x_{v} = - \frac{180}{2 (- 2)} x_{v} = 45$

Substitute $x_{v} = 45$ in the original equation to find $y_{v} :$

$y_{v} = 4050 Therefore 鈥� the 鈥� parabola 鈥� vertex 鈥� is (45, 鈥� 4050) If 鈥� a < 0, 鈥� then 鈥� the 鈥� vertex 鈥� is 鈥� a 鈥� maximum 鈥� value If 鈥� a > 0, 鈥� then 鈥� the 鈥� vertex 鈥� is 鈥� a 鈥� minimum 鈥� value a = - 2 Maximum (45, 鈥� 4050)$

Therefore, the length of the building that should border the yard to maximize the area is $45$ ft. The maximum area is $4050 {ft}^{2}$ .

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91影视

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the maximum area.

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