Answer:
x = 2√30
y= 3√30
Step-by-step explanation:
Area= 180in^2
For the larger shape,
Let the length be x and the width be y
A = x. y
x. y = =180
y= 180/x
For the smaller shape, lenght= x-2 while width = y-3
Area = (x-2)(y-3)
A(x) = (x-2)(y-3)
Put y= 180/x into A(x)
= (x-2)(180/x -3)
= 180 - 3x - 360/x + 6
Differentiate A(x) with respect to x
A'(x) = -3x+ 360/x^2
A'(x) = 0
0 = -3x + 360/x^2
3x = 360/x^2
3x^3 = 360
x^3 = 360/3
x^3 = 120
x = cuberoot 120
x = 2√30
Recall that y= 180/x
y = 180/2√30
y= 90/√30
By rationalizing the surd, we have
y = 90/√30 * √30/√30
y = (90√30) /30
y = 3√30
x = 2√30, y = 3√30
The area of the poster and the dimensions of the margins are given, while the dimensions (height and width) of the poster are to be determined
The dimensions of the poster that gives the largest printed area are;
The width of the poster is 2·√(30) inches
The height of the poster is 3·√30 inches
Reason:
The given parameter are;
The area of the poster = 180 in.²
Margins at the bottom and sides = 1 inch
The margin at the top = 2 inches
Required;
The dimensions of the poster that gives the largest print area.
Solution;
Let h represent the height of the poster, and let w represent the width of
the poster, we have;
Area of the poster, A = h × w = 186
Printed area, [tex]A_{pri}[/tex] = (w - 2) × (h - 3)
Therefore, we get;
[tex]A_{pri} = (w - 2) \times \left(\dfrac{180}{w} - 3\right) = -\dfrac{3\cdot w^2-186 \cdot w + 360}{w}[/tex]
The leading coefficient of the function for printed area is negative,
therefore, the function has a maximum point given as follows;
[tex]\dfrac{dA_{pri} }{dw} = \dfrac{d}{dw} \left(-\dfrac{3\cdot w^2-186 \cdot w + 360}{w} \right) = 0[/tex]
Which gives;
[tex]\dfrac{d}{dw} \left(-\dfrac{3\cdot w^2-186 \cdot w + 360}{w} \right) =-\dfrac{3\cdot w^2-360}{w^2}[/tex]
[tex]w^2 = \dfrac{360}{3} = 120[/tex]
w = ±2·√30
Therefore, the width of the poster, w = 2·√(30) inches
[tex]The \ height \ of \ the \ poster, \ h = \dfrac{180}{2\cdot \sqrt{30} } = 3 \cdot \sqrt{30}[/tex]
The height of the poster, h = 3·√30 inches
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