The height of a triangle is 3 inches less than 5 times the length of its base. The area is 34 square inches. Find the length of the base and the height of the triangle.
The subject is not San Bernadino valley College.
Let L = length
and h = height
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h = 5L-3
1/2(hL) = 34
Solve for h and L.
Substitute the value of h in the first equation into the second equation as follows:
1/2(5L-3)*L = 34
Now solve for L, then h.
Drag the yellow point until an accurate height of the triangle is drawn afterwards fill out empty boxes below to determine the area of the triangle
To find the length of the base and the height of the triangle, we can first write an equation using the given information.
Let's assume the length of the base is represented by 'b' inches. According to the problem, the height of the triangle is 3 inches less than 5 times the length of the base. This can be written as:
height = 5b - 3 inches
The formula for the area of a triangle is given by:
area = (1/2) * base * height
We are given that the area is 34 square inches. Substituting the values into the equation:
34 = (1/2) * b * (5b - 3)
Now we can solve this equation to find the value of 'b', which represents the length of the base. To do this, we will first simplify the equation:
34 = (1/2) * (5b^2 - 3b)
Multiplying both sides by 2 to eliminate the fraction:
68 = 5b^2 - 3b
Rearranging the equation to form a quadratic equation in standard form:
5b^2 - 3b - 68 = 0
This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. Once we find the value of 'b', we can substitute it back into the expression for the height to find the height of the triangle.
Let's solve the quadratic equation using the quadratic formula:
b = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 5, b = -3, and c = -68. Substituting these values into the formula:
b = (-(-3) ± √((-3)^2 - 4 * 5 * -68)) / (2 * 5)
= (3 ± √(9 + 1360)) / 10
= (3 ± √1369) / 10
Since we are dealing with lengths, we discard the negative root and simplify the square root:
b = (3 + √1369) / 10
Now we can calculate the value of 'b':
b = (3 + √1369) / 10
≈ (3 + 37) / 10
≈ 40 / 10
= 4 inches
So, the length of the base is 4 inches.
To find the height of the triangle, we can substitute this value back into the expression for the height:
height = 5b - 3
= 5(4) - 3
= 20 - 3
= 17 inches
Therefore, the length of the base is 4 inches and the height of the triangle is 17 inches.