Question

What is the expression?

for the base of the triangle?

The base of a triangle is four less than three times the height. The area of the triangle is 38 square centimeters.

Answer 1

To express the base of the triangle in terms of the height, we can define variables for height and base:

Let \( h \) be the height of the triangle. According to the problem, the base \( b \) can be expressed as:

\[
b = 3h - 4
\]

Next, we know that the area \( A \) of a triangle can be calculated using the formula:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the known area and the expression for the base into this formula, we have:

\[
38 = \frac{1}{2} \times (3h - 4) \times h
\]

Now, we can simplify this equation:

1. Multiply both sides by 2 to eliminate the fraction:

\[
76 = (3h - 4) \times h
\]

2. Distribute \( h \):

\[
76 = 3h^2 - 4h
\]

3. Rearranging the equation, we get a standard quadratic form:

\[
3h^2 - 4h - 76 = 0
\]

This quadratic equation can be solved for \( h \) using the quadratic formula:

\[
h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 3 \), \( b = -4 \), and \( c = -76 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-4)^2 - 4 \times 3 \times (-76) = 16 + 912 = 928
\]

Now, proceeding with the quadratic formula:

\[
h = \frac{4 \pm \sqrt{928}}{6}
\]

Calculating \( \sqrt{928} \) yields approximately 30.4. So:

\[
h = \frac{4 \pm 30.4}{6}
\]

This gives us two solutions:

1. \( h = \frac{34.4}{6} \approx 5.73 \)
2. \( h = \frac{-26.4}{6} \) (we discard this since height cannot be negative)

Finally, we substitute this height back into our expression for the base:

\[
b = 3(5.73) - 4 \approx 17.19 - 4 = 13.19
\]

Thus, the base of the triangle \( b \) is approximately \( 13.19 \) cm.