Question

the area of a triangle can be represented by the expression (14x^{6}+63x^{2}). if the base is (7x^{2}), write an expression to represent its height. ((a = \frac{1}{2}bh))

Explanation:

Step1: Recall the area formula of a triangle

The formula for the area of a triangle is \( A=\frac{1}{2}bh \), where \( A \) is the area, \( b \) is the base, and \( h \) is the height. We need to solve for \( h \), so we can rearrange the formula. First, multiply both sides of the equation by 2 to get rid of the fraction: \( 2A = bh \). Then, divide both sides by \( b \) to solve for \( h \), so \( h=\frac{2A}{b} \).

Step2: Substitute the given expressions for \( A \) and \( b \)

We are given that \( A = 14x^{6}+63x^{2} \) and \( b = 7x^{2} \). Substitute these into the formula for \( h \):
\( h=\frac{2(14x^{6}+63x^{2})}{7x^{2}} \)

Step3: Simplify the numerator

First, distribute the 2 in the numerator: \( 2(14x^{6}+63x^{2})=28x^{6}+126x^{2} \). So now we have \( h=\frac{28x^{6}+126x^{2}}{7x^{2}} \)

Step4: Divide each term in the numerator by the denominator

We can split the fraction into two separate fractions: \( h=\frac{28x^{6}}{7x^{2}}+\frac{126x^{2}}{7x^{2}} \)

Step5: Simplify each fraction

For the first fraction, \( \frac{28x^{6}}{7x^{2}} \), divide the coefficients: \( \frac{28}{7} = 4 \), and use the quotient rule for exponents \( \frac{x^{m}}{x^{n}}=x^{m - n} \), so \( x^{6-2}=x^{4} \). So the first term simplifies to \( 4x^{4} \).

For the second fraction, \( \frac{126x^{2}}{7x^{2}} \), divide the coefficients: \( \frac{126}{7}=18 \), and \( \frac{x^{2}}{x^{2}} = 1 \) (since any non - zero number to the power of 0 is 1, and \( x^{2-2}=x^{0}=1 \) for \( x
eq0 \)). So the second term simplifies to 18.

Putting it together, \( h = 4x^{4}+18 \)

Answer:

The expression for the height of the triangle is \( 4x^{4}+18 \)