Final answer:
The problem is solved by setting up an equation using the area of a triangle formula, which then is manipulated to derive a quadratic equation. The positive solution for the base is then used to determine the height, which is five inches less than the base.
Step-by-step explanation:
Finding the Base and Height of a Triangle Given Its Area
To solve the design problem where the height of a triangle is five inches less than the length of its base, and the area of the triangle is 52 square inches, we can use the formula for the area of a triangle which is Area = 1/2 × base × height.
Let's define the base of the triangle as b inches. According to the problem, the height (h) will then be b - 5. Plugging these values into the formula for the area, we get:
52 = 1/2 × b × (b - 5)
To solve for b, we first multiply both sides by 2 to get rid of the fraction:
104 = b × (b - 5)
Next, distribute b across the expression (b - 5):
104 = b^2 - 5b
This leaves us with a quadratic equation. Setting it equal to zero:
b^2 - 5b - 104 = 0
We then factor or use the quadratic formula to find the value of b. Assuming we get a positive value for the base, we plug it back into h = b - 5 to find the height.
Following these steps will provide the dimensions of the base and height of the triangle.