A rectangle is to be inscribed in a right triangle having sides of length 36 in, 48 in, and 60 in. Find the dimensions of the rectangle with greatest area assuming the rectangle is positioned as in the accompanying figure.
right triangle: hypotenuse- 60in
base-36in
height-48in
hell mon
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To find the dimensions of the rectangle with the greatest area inscribed in the right triangle, we can start by analyzing the properties of this problem.
Since the rectangle is inscribed in the right triangle, its sides will be parallel to the sides of the right triangle. Let's assume the sides of the rectangle have lengths a and b, with a being the length parallel to the base of the right triangle (36 in) and b being the length parallel to the height of the right triangle (48 in).
To find the dimensions of the rectangle with the greatest area, we need to maximize the area of the rectangle.
The area of a rectangle is given by the formula A = length * width. In this case, the area of the rectangle would be A = a * b.
Now, since the sides of the rectangle are parallel to the sides of the right triangle, we can observe that the length a is equal to the base of the right triangle (36 in) minus the width of the rectangle. Similarly, the length b is equal to the height of the right triangle (48 in) minus the height of the rectangle.
Using this information, we can express the area A in terms of the width and height of the rectangle:
A = (36 - width) * (48 - height)
To find the dimensions of the rectangle that maximize the area, we need to find the values of width and height that maximize the expression (36 - width) * (48 - height).
To do this, we can take the partial derivatives of A with respect to both width and height, set them equal to zero and solve for width and height.
∂A/∂width = -1 * (48 - height) = 0
∂A/∂height = (36 - width) * (-1) = 0
Solving these equations, we get:
48 - height = 0, which results in height = 48
36 - width = 0, which results in width = 36
Therefore, the dimensions of the rectangle with the greatest area that can be inscribed in the given right triangle are:
Width = 36 in
Height = 48 in