Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 90 feet of fencing to go around it, what dimensions will give him the maximum area in the garden.

Step-by-step explanation:

Let the rectangle dimensions be x and y. Assume that x represents the "short" side, so it is also the diameter of the semicircle. Then the perimeter is ...

  P = x + 2y + πx/2 = 90

The area is the area of the rectangle added to the area of the semicircle:

  A = xy + (1/2)π(x/2)^2 = x(π/8x +y)

Solving the perimeter equation for y, we get

  y = (90 -x -π/2x)/2

and the area equation becomes ,..

  A = x(π/8x +45 -x/2 -π/4x)

  A = x(45 -x(1/2 +π/8))

The graph of A(x) is a parabola opening downward with zeros at x=0 and x=45/(1/2+π/8) = 360/(4+π).

The vertex (maximum) of the area curve will be halfway between the zeros, at ...

  x = (0 +360/(4+π))/2 = 180/(4+π)

The value of y is then ...

  y = (90 -x(1 +π/2))/2 = (1/2)(90 -(180/(4+π))(2+π)/2) = (1/2)(90)(1 -(2+π)/(4+π))

  y = 45(4+π-2-π)/(4+π) = 90/(4+π)

The rectangle dimensions are 90/4+π) by 180/(4+π), with the semicircle diameter equal to 180/(4+π).

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That's about 25.2 ft by 12.6 ft with the semicircle attached to the long side.

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Please note that the overall dimensions of the garden are the same in both directions. It is effectively "square" with two corners rounded into quarter circles.

the length of o'p' is .

step-by-step explanation:

it is given that the quadrilateral m'n'o'p' is the image of quadrilateral mnop after dilation.

the center of dilation is origin and the scale factor is 1.5.

in dilation the image and preimge are similar figures. it means the sides of image and preimage are proportional.

the scale factor is defined as

the scale factor is 1.5.

multiply both sides by op.

note: if the value of op is given, then multiply the value of op with 1.5 to get the length of side o'p'.

therefore length of o'p' is .

#12 the answer is the third option