Does anyone know how to do this question? The cross section of the roof of a house is modelled by the function y= -5/12|x-12|+5 I need to find the slope, height,and length of both sides including the base of this isosceles triangle. I'm having a extremely difficult time with this please help if you can any help at all would be greatly appreciated.
To find the slope, height, and length of both sides including the base of the isosceles triangle formed by the cross-section of the roof, you can follow these steps:
Step 1: Understand the Equation
First, let's understand the equation of the cross-section function, y = -5/12|x-12| + 5. This equation represents a V-shape roof with its vertex at (12, 5). The absolute value |x-12| ensures that both sides of the roof are symmetric.
Step 2: Identify the Important Points
To find the slope, height, and length of the sides and base, we need to identify the key points in the equation. These key points are the vertex and the x-intercepts.
The vertex of the function is given by (12, 5), which represents the highest point of the roof.
To find the x-intercepts, set y = 0 and solve for x:
-5/12|x-12| + 5 = 0
-5/12|x-12| = -5
|x-12| = 12
x-12 = 12 or x-12 = -12
Solving the two equations, we get:
x = 24 or x = 0
Therefore, the x-intercepts are at (0, 0) and (24, 0), which represent the points where the roof intersects the x-axis.
Step 3: Calculate the Slope
The slope of the roof can be found using the formula: slope = (change in y) / (change in x).
To calculate the slope of each side, we need to find the change in y and change in x between the vertex and the x-intercepts.
For the left side:
Change in y = 0 - 5 = -5
Change in x = 12 - 0 = 12
For the right side:
Change in y = 0 - 5 = -5
Change in x = 24 - 12 = 12
Hence, the slope of both sides is -5/12.
Step 4: Compute the Height
The height of the isosceles triangle is the distance between the vertex (12, 5) and the base of the triangle (x-intercepts).
Using the distance formula: height = √((x2-x1)^2 + (y2-y1)^2), we can calculate the height for each side.
For the left side:
Height = √((0-12)^2 + (0-5)^2)
= √((-(12))^2 + (-5)^2)
= √(144 + 25)
= √169
= 13
For the right side:
Height = √((24-12)^2 + (0-5)^2)
= √(12^2 + (-5)^2)
= √(144 + 25)
= √169
= 13
Hence, the height of both sides is 13.
Step 5: Determine the Length of the Base
The length of the base of the triangle corresponds to the distance between the x-intercepts (x = 0) and (x = 24).
Length of the base = 24 - 0 = 24.
So, the length of the base is 24.
To summarize:
- The slope of both sides is -5/12.
- The height of both sides is 13.
- The length of the base is 24.