Question
A triangle with an area of 224 square centimeters has a height that is 8 more than 6 times its base. Find the base and height, in centimeters, of the triangle.
Answers
GPT-4o mini
To solve the problem, we need to use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Given that the area of the triangle is \(224\) square centimeters, we can set up the equation:
\[
\frac{1}{2} \times b \times h = 224
\]
Multiplying both sides by \(2\), we get:
\[
b \times h = 448
\]
Next, we are told that the height \(h\) is 8 more than 6 times the base \(b\). We can express this relationship as:
\[
h = 6b + 8
\]
Now, we can substitute the expression for \(h\) into the area equation:
\[
b \times (6b + 8) = 448
\]
This expands to:
\[
6b^2 + 8b = 448
\]
To turn it into a standard quadratic equation, we'll subtract \(448\) from both sides:
\[
6b^2 + 8b - 448 = 0
\]
Now, we can simplify this equation by dividing everything by \(2\):
\[
3b^2 + 4b - 224 = 0
\]
Next, we will solve this quadratic equation using the quadratic formula:
\[
b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]
In our equation, \(A = 3\), \(B = 4\), and \(C = -224\). First, we calculate the discriminant:
\[
B^2 - 4AC = 4^2 - 4 \times 3 \times (-224) = 16 + 2688 = 2704
\]
Now we find the square root of the discriminant:
\[
\sqrt{2704} = 52
\]
Now we can apply the quadratic formula:
\[
b = \frac{-4 \pm 52}{2 \times 3} = \frac{-4 \pm 52}{6}
\]
This gives us two possible values for \(b\):
1. When we use the positive root:
\[
b = \frac{48}{6} = 8
\]
2. When we use the negative root:
\[
b = \frac{-56}{6} = -\frac{28}{3} \quad (\text{not a valid solution as base cannot be negative})
\]
Thus, we have:
\[
b = 8 \text{ cm}
\]
Now we can find \(h\) using the relationship \(h = 6b + 8\):
\[
h = 6(8) + 8 = 48 + 8 = 56 \text{ cm}
\]
Finally, the base and height of the triangle are:
\[
\text{Base} = 8 \text{ cm}, \quad \text{Height} = 56 \text{ cm}
\]
Thus, the solution is:
\[
\boxed{8} \text{ cm for the base and } \boxed{56} \text{ cm for the height.}
\]
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Given that the area of the triangle is \(224\) square centimeters, we can set up the equation:
\[
\frac{1}{2} \times b \times h = 224
\]
Multiplying both sides by \(2\), we get:
\[
b \times h = 448
\]
Next, we are told that the height \(h\) is 8 more than 6 times the base \(b\). We can express this relationship as:
\[
h = 6b + 8
\]
Now, we can substitute the expression for \(h\) into the area equation:
\[
b \times (6b + 8) = 448
\]
This expands to:
\[
6b^2 + 8b = 448
\]
To turn it into a standard quadratic equation, we'll subtract \(448\) from both sides:
\[
6b^2 + 8b - 448 = 0
\]
Now, we can simplify this equation by dividing everything by \(2\):
\[
3b^2 + 4b - 224 = 0
\]
Next, we will solve this quadratic equation using the quadratic formula:
\[
b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
\]
In our equation, \(A = 3\), \(B = 4\), and \(C = -224\). First, we calculate the discriminant:
\[
B^2 - 4AC = 4^2 - 4 \times 3 \times (-224) = 16 + 2688 = 2704
\]
Now we find the square root of the discriminant:
\[
\sqrt{2704} = 52
\]
Now we can apply the quadratic formula:
\[
b = \frac{-4 \pm 52}{2 \times 3} = \frac{-4 \pm 52}{6}
\]
This gives us two possible values for \(b\):
1. When we use the positive root:
\[
b = \frac{48}{6} = 8
\]
2. When we use the negative root:
\[
b = \frac{-56}{6} = -\frac{28}{3} \quad (\text{not a valid solution as base cannot be negative})
\]
Thus, we have:
\[
b = 8 \text{ cm}
\]
Now we can find \(h\) using the relationship \(h = 6b + 8\):
\[
h = 6(8) + 8 = 48 + 8 = 56 \text{ cm}
\]
Finally, the base and height of the triangle are:
\[
\text{Base} = 8 \text{ cm}, \quad \text{Height} = 56 \text{ cm}
\]
Thus, the solution is:
\[
\boxed{8} \text{ cm for the base and } \boxed{56} \text{ cm for the height.}
\]