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Chapter 2: Problem 9

Solve each formula for the specified variable. \(\mathscr{A}=\frac{1}{2} h(b+B)\) (area of a trapezoid) (a) for \(h\) (b) for \(B\)

Short Answer

Expert verified
(a) \( h = \frac{2 \mathscr{A}}{b + B} \)(b) \( B = \frac{2 \mathscr{A}}{h} - b \)

Step by step solution

01

- Write Down the Given Formula

The formula given is the area of a trapezoid: equation\[ \text{} \ \ \mathscr{A} = \frac{1}{2} h(b + B) \text{} \] Our goal is to solve this formula for two different variables: (a) for \(h\) and (b) for \(B\).
02

- Solve for \( h \)

To isolate \( h \), start by multiplying both sides of the equation by 2 to get rid of the fraction: equation\[ 2 \mathscr{A} = h(b + B) \] Now divide both sides by \( (b + B) \): equation\[ h = \frac{2 \mathscr{A}}{b + B} \] This is the formula solved for \( h \).
03

- Solve for \( B \)

To isolate \( B \), start by multiplying both sides of the original equation by 2 again: equation\[ 2 \mathscr{A} = h(b + B) \] Next, divide both sides by \( h \): equation\[ \frac{2 \mathscr{A}}{h} = b + B \] Finally, subtract \( b \) from both sides to solve for \( B \): equation\[ B = \frac{2 \mathscr{A}}{h} - b \] This is the formula solved for \( B \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formula Manipulation
Formula manipulation is an essential skill in algebra. It involves rearranging and rewriting equations to isolate a specific variable. This skill allows you to solve for unknowns in a variety of mathematical and scientific contexts. To manipulate a formula effectively, you must perform the same operations on both sides of the equation; this keeps the equation balanced. For instance, multiplying both sides of an equation by 2 undoes division by 2, and vice versa.
When solving equations, you may need to use operations like:
  • Adding or subtracting terms
  • Multiplying or dividing by constants or variables
  • Using parentheses to group terms appropriately
Let's see these principles in action with the provided formula for the area of a trapezoid.
Area of a Trapezoid
The area of a trapezoid is calculated using the formula: \(\text{} \ \mathscr{A} = \frac{1}{2} h(b + B) \ \). This formula is derived from the average of the lengths of the two parallel sides (often called the bases, labeled as 'b' and 'B') and the height 'h'. Understanding this formula helps in geometric calculations and is necessary for solving relevant algebra problems.
Here's a quick explanation of each term:
  • \( \mathscr{A} \) is the area of the trapezoid.
  • 'h' is the height, which is the perpendicular distance between the two parallel sides.
  • 'b' and 'B' are the lengths of the two parallel sides.
By manipulating this formula, you can solve for any of these variables if you know the others. This flexibility is incredibly useful in both pure math and applied sciences.
Isolating Variables
Isolating variables is a standard method in algebra to solve for a specific variable. You 'isolate' a variable by performing operations that get the variable alone on one side of the equation. Let's apply this method to solve for both 'h' and 'B' in the trapezoid area formula.
First, to isolate 'h':
  • Multiply both sides by 2: \(\text{} \ 2 \mathscr{A} = h(b + B) \ \)
  • Divide both sides by \( b + B \): \(\text{} \ h = \frac{2 \mathscr{A}}{b + B} \ \)

Now to isolate 'B':
  • Again, start by multiplying both sides by 2: \(\text{} \ 2 \mathscr{A} = h(b + B) \ \)
  • Next, divide both sides by 'h': \( \ \frac{2 \mathscr{A}}{h} = b + B \)
  • Finally, subtract 'b' from both sides: \( \ B = \frac{2 \mathscr{A}}{h} - b \)
By mastering these steps, you gain the ability to rearrange and solve various formulas for any variable you need.

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Most popular questions from this chapter

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) Sondra Braeseker bought a new car and agreed to pay it off in 36 monthly payments. The total finance charge was \(\$ 700 .\) Find the unearned interest if she paid the loan off 4 payments ahead of schedule.

Evaluate. Half of \(-18,\) added to the reciprocal of \(\frac{1}{5}\)

How much pure dye must be added to 4 gal of a \(25 \%\) dye solution to increase the solution to \(40 \% ?\) (Hint: Pure dye is \(100 \%\) dye. \()\)

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) The finance charge on a loan taken out by Kha Le is \(\$ 380.50 .\) If 24 equal monthly installments were needed to repay the loan, and the loan is paid in full with 8 months remaining, find the amount of unearned interest.

Express each set in the simplest interval form. $$ [-1, \infty) \cap(-\infty, 9] $$
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