Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A=\frac{1}{2} h(a+b) \text { for } b$$

Step-by-Step Solution

Answer

The solution for variable b is \(b = \frac{2A - ah}{h}\).

Step 1: Understand the Problem

The problem is to solve the formula \(A=\frac{1}{2} h(a+b)\) for b. This is the area of a trapezoid special formula. Area A, height h, and the lengths of the two parallel sides a and b are given. We need to isolate b in this equation.

Step 2: Isolate the term with b

To isolate b, the first step is to eliminate the fraction by multiplying both sides of the equation by 2. This gives us \(2A = h(a + b)\).

Step 3: Distribute h across the brackets

Distributing h across (a + b) will not change the equation but will bring us closer to isolating b. After the distribution, the equation becomes \(2A = ah + bh\).

Step 4: Move ah to the other side

Subtract ah from both sides of the equation to isolate the term with b. This gives us \(2A - ah = bh\).

Step 5: Solve for b

Finally, divide both sides of the equation by h to solve for b. So, the solution for b is \(b = \frac{2A - ah}{h}\).