Sure, let's work through the problem step-by-step.
### Part (a) - Solving for [tex]\( b \)[/tex] when [tex]\( A = 390 \)[/tex] and [tex]\( h = 26 \)[/tex]:
We start with the formula for the area of a triangle:
[tex]\[ A = \frac{1}{2} b h \][/tex]
Given:
- [tex]\( A = 390 \)[/tex]
- [tex]\( h = 26 \)[/tex]
We need to solve for the base [tex]\( b \)[/tex].
First, substitute the given values of [tex]\( A \)[/tex] and [tex]\( h \)[/tex] into the formula:
[tex]\[ 390 = \frac{1}{2} b \cdot 26 \][/tex]
Next, isolate [tex]\( b \)[/tex] by performing the following steps:
1. Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
2 \cdot 390 = b \cdot 26
\][/tex]
[tex]\[
780 = b \cdot 26
\][/tex]
2. Divide both sides by 26 to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \frac{780}{26}
\][/tex]
[tex]\[
b = 30
\][/tex]
So, when [tex]\( A = 390 \)[/tex] and [tex]\( h = 26 \)[/tex], the base [tex]\( b \)[/tex] is [tex]\( 30 \)[/tex].
### Part (b) - Solving for [tex]\( b \)[/tex] in general:
To find the general formula for [tex]\( b \)[/tex], start from the area formula:
[tex]\[ A = \frac{1}{2} b h \][/tex]
We need to solve for [tex]\( b \)[/tex]. Rearrange the equation to isolate [tex]\( b \)[/tex]:
[tex]\[ \frac{1}{2} b h = A \][/tex]
First, multiply both sides by 2 to get rid of the fraction:
[tex]\[ b h = 2A \][/tex]
Then, divide both sides by [tex]\( h \)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \frac{2A}{h} \][/tex]
This gives us the general formula for [tex]\( b \)[/tex] in terms of [tex]\( A \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ b = \frac{2A}{h} \][/tex]
Therefore, in general, the base [tex]\( b \)[/tex] can be calculated using the formula:
[tex]\[ b = \frac{2A}{h} \][/tex]
