A window has the shape of a square surmounted by an isosceles triangle. The sides of the square are length x. The The perimeter of the window is 90 inches. Thus, the perimeter of the triangle is x+2y. So, 90 inches = 3x+2y.
(a) Write an equation for the area of the window in terms of x and y.
(b) Use the fact that the perimeter of the window is 90 inches to write the area as a function
of only x.
(c) In this situation, what is the largest that x can be? What is the smallest x can be?
THANK YOU FOR ANY AND ALL HELP!
To answer these questions, let's break down the problem step by step:
(a) To find the equation for the area of the window in terms of x and y, we need to determine the individual areas of the square and the triangle.
The area of the square is calculated by squaring the length of one side, so it's x^2.
The area of the isosceles triangle is given by the formula: A = (1/2) base * height.
In this case, the base is x, and the height can be found by using the Pythagorean theorem. The two equal sides of the triangle are also x because they correspond to the sides of the square. The hypotenuse of the triangle (which is the height of the triangle) can be found by using y, as mentioned in the problem description. Applying the Pythagorean theorem, we have y^2 = (x/2)^2 + x^2, which simplifies to y^2 = (1/4)x^2 + x^2 = (5/4)x^2. Taking the square root of both sides, we get y = √((5/4)x^2) = √(5/4)x.
Now we can calculate the area of the triangle:
A_triangle = (1/2) * base * height = (1/2) * x * √(5/4)x = (1/2) * √5/2 * x^(3/2) = √5/8 * x^(3/2).
The total area of the window is the sum of the squares and the triangle:
A = A_square + A_triangle = x^2 + √5/8 * x^(3/2).
(b) Next, let's express the area as a function of x only using the fact that the perimeter of the window is 90 inches. From the problem description, we know that the perimeter is given by 3x + 2y = 90.
Solving this equation for y, we find:
2y = 90 - 3x, and dividing both sides by 2, we get:
y = (90 - 3x) / 2.
Substituting this value of y into our equation for the area, we obtain:
A = x^2 + √5/8 * x^(3/2) = x^2 + √5/8 * x^(3/2) = x^2 + √5/8 * (90 - 3x)^(3/2).
Thus, the area of the window can be expressed as a function of x only as:
A(x) = x^2 + √5/8 * (90 - 3x)^(3/2).
(c) Finally, to determine the largest and smallest possible values for x, we can consider the constraints on the problem.
The length of a side, x, cannot be negative in this context, so x > 0.
Also, the triangle is surmounting the square, so the length of the triangle's base (which is also the side length of the square) must be greater than zero, x > 0.
From the perimeter equation 3x + 2y = 90, we can isolate y:
2y = 90 - 3x,
y = (90 - 3x)/2.
Since y is the height of the triangle, it must also be greater than zero, y > 0.
From this inequality, we can conclude:
(90 - 3x)/2 > 0,
90 - 3x > 0,
-3x > -90,
x < 30.
Therefore, the largest value x can take is 30.
In conclusion, x can vary from 0 to 30.