Question

the triangle shown below has an area of 37n^3 + 17n^2 + 11n. find a simplified polynomial expression for its height.

Explanation:

Step1: Recall triangle - area formula

The area formula of a triangle is $A=\frac{1}{2}bh$, where $A$ is the area, $b$ is the base, and $h$ is the height. We know that $A = 37n^{3}+17n^{2}+11n$ and $b = n$.

Step2: Solve for height $h$

We can rewrite the area formula to solve for $h$: $h=\frac{2A}{b}$. Substitute $A = 37n^{3}+17n^{2}+11n$ and $b = n$ into the formula.
\[

$$\begin{align*} h&=\frac{2(37n^{3}+17n^{2}+11n)}{n}\\ &=\frac{2\times37n^{3}+2\times17n^{2}+2\times11n}{n}\\ &=\frac{74n^{3}+34n^{2}+22n}{n} \end{align*}$$

\]

Step3: Simplify the expression

Using the rule $\frac{a + b + c}{d}=\frac{a}{d}+\frac{b}{d}+\frac{c}{d}$, we have:
\[

$$\begin{align*} h&=\frac{74n^{3}}{n}+\frac{34n^{2}}{n}+\frac{22n}{n}\\ &=74n^{2}+34n + 22 \end{align*}$$

\]

Answer:

$74n^{2}+34n + 22$