"The height of a triangle is 4 ft more than its base. Its area is 70 square feet. Find the height and base, then enter their sum below."
Also, there is this one^ I'm just really confused about how to set it up.
b = base
h = height
Area of triangle:
A = b ∙ h / 2
In this case:
h = b + 4 , A = 70 ft²
A = b ∙ h / 2
70 = b ∙ ( b + 4 ) / 2
70 = ( b² + 4 b ) / 2
Multiply both sides by 2
140 = b² + 4 b
Subtract 140 to both sides
0 = b² + 4 b - 140
b² + 4 b - 140 = 0
The solutions are:
b = - 14 and b = 10
Base can not be negative so:
b = 10 ft
h = b + 4
h = 14 ft
Thank you!! I'm a little confused on what the sum would be, would it be 4?
The height of a triangle is 4 ft more than its base means:
The height
h = base + 4
base + 4 is 4 ft more than its base
"Their sum" refers to the result you would get by adding them together.
Since the base is 10 and the height is 14, their sum is 10 +14 = 24
To solve this problem, you can use the formula for the area of a triangle, which is given by "Area = (1/2) * base * height".
Let's break down the problem and set up the equations:
1. "The height of a triangle is 4 ft more than its base."
Let's assume the base of the triangle is "x" ft.
So, the height of the triangle would be "x + 4" ft.
2. "Its area is 70 square feet."
The area of the triangle is given as 70 square feet.
Now, we can substitute these values into the formula for the area to create an equation:
Area = (1/2) * base * height
70 = (1/2) * x * (x + 4)
By simplifying the equation, we can find the value of "x" (the base of the triangle). Once we have the base, we can find the height by substituting the value of "x" into the expression "x + 4".
To solve the equation for "x", let's continue with the algebraic steps:
70 = (1/2) * x * (x + 4)
Multiply both sides of the equation by 2 to eliminate the fraction:
140 = x * (x + 4)
Distribute x to the terms inside the parenthesis:
140 = x^2 + 4x
Rearrange the equation to set it equal to zero:
x^2 + 4x - 140 = 0
Now, we have a quadratic equation. Factoring this equation may not be straightforward, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 4, and c = -140. Substituting these values into the quadratic formula:
x = (-(4) ± √((4)^2 - 4(1)(-140))) / (2(1))
After simplifying:
x = (-4 ± √(16 + 560)) / 2
x = (-4 ± √576) / 2
x = (-4 ± 24) / 2
By evaluating both possibilities, we have two potential values for x:
1. x = (-4 + 24) / 2 = 20 / 2 = 10
2. x = (-4 - 24) / 2 = -28 / 2 = -14
Since we are dealing with the side lengths of a triangle, negative values do not make sense in this context. Therefore, we discard the negative value.
Hence, the base of the triangle is 10 ft.
To find the height, substitute the value of x into "x + 4":
Height = 10 + 4 = 14 ft.
Finally, to find the sum of the base and height:
Sum = Base + Height = 10 + 14 = 24 ft.
Therefore, the sum of the base and height is 24 ft.