An architect is designing a small A-frame cottage for a new resort area. A cross section of the cottage is an isosceles triangle with an area of 98 square feet. The front wall of the cottage must a accommodate a sliding door that is 6 feet wide and 8 feet high. Find the width and height of the cross section
center the door way so that half the base is 3+x feet. The area of the triangle is 98, so the height h = 98/(3+x)
using similar triangles,
8/x = h/(x+3)
8(x+3) = hx
8(x+3) = 98/(x+3) * x
4x^2 - 25x + 36 = 0
(4x-9)(x-4) = 0
Looks like any x from 9/4 to 4 will work. So, the dimensions can vary from
10.5 ft wide x 18 2/3 ft high
to
14 wide x 16 ft high
To find the width and height of the cross section, we can use the formula for the area of a triangle.
Let's assume the width of the cross-section is the base of the isosceles triangle, and the height is the height of the triangle.
The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
Given that the area of the triangle is 98 square feet, we can set up the equation as follows:
98 = (1/2) * base * height
Since the front wall of the cottage must accommodate a sliding door that is 6 feet wide and 8 feet high, the base and height of the cross-section should be greater than or equal to these dimensions.
Let's set the base as the width of the sliding door, which is 6 feet, and the height as the height of the sliding door, which is 8 feet.
Substituting these values into the equation, we get:
98 = (1/2) * 6 * 8
Simplifying the equation, we have:
98 = 24
However, this is not a valid equation as it does not hold true, which means our assumption was incorrect.
In this case, we need to consider that the width and height of the cross-section are greater than the dimensions of the sliding door.
Let's assume the base of the triangle is x feet, and the height of the triangle is y feet.
Now we can set up the equation again:
98 = (1/2) * x * y
Since the front wall of the cottage must accommodate a sliding door that is 6 feet wide and 8 feet high, we have the following inequalities:
x >= 6
y >= 8
Let's solve for x and y.
By rearranging the equation, we can isolate x:
98 * 2 = x * y
196 = x * y
x = 196 / y
Substituting this expression for x into the inequality, we have:
196 / y >= 6
Simplifying the inequality, we find:
y <= 196 / 6
y <= 32.67
Since the height of the triangle must be greater than or equal to 8 feet, the height is 8 feet.
Now, let's substitute this value into the equation to solve for x:
98 = (1/2) * x * 8
Simplifying the equation:
98 = 4x
Dividing both sides by 4:
24.5 = x
Therefore, the width of the cross-section is approximately 24.5 feet, and the height is 8 feet.
To find the width and height of the cross section, we can use the formula for the area of a triangle, which is (base x height) / 2.
Let's assume the base of the triangle is the width of the cottage. So, the area of the triangle is (width x height) / 2.
We are given that the area of the triangle is 98 square feet. Therefore, we have the equation:
(width x height) / 2 = 98
Now let's consider the sliding door on the front wall. The sliding door has a width of 6 feet and a height of 8 feet. The base of the triangle, which is the width of the cottage, should be larger than the door's width to accommodate it.
So, let's set up a system of equations to solve for the width and height of the cross section.
Equation 1: (width x height) / 2 = 98
Equation 2: width > 6 (to accommodate the sliding door)
To solve this system of equations, we need to find a possible value for the width (base), and then calculate the corresponding height.
Let's start with the assumption that the width is 7 feet (greater than 6 to accommodate the door). Plugging this into Equation 1:
(7 x height) / 2 = 98
Next, we isolate the height by multiplying both sides by 2 and dividing by 7:
height = (98 x 2) / 7
height ≈ 28
So, for a width of 7 feet, the corresponding height is approximately 28 feet.
Now, let's check if this solution satisfies Equation 2 (width > 6):
7 > 6
Since this is true, the width of 7 feet and height of approximately 28 feet satisfy both equations.
Therefore, the width of the cross section is 7 feet, and the height is approximately 28 feet.