Chapter 1 Equations, Inequalities, and Mathematical Models 1.3 Formulas and Applications

Objectives At the end of this session, you will be able to:  Solve problems using formulas.  Use linear equations to solve problems.  Solve for a specified variable in a formula.

Contents 1. Formulas 2. Problem Solving with Linear Equations 3. Solving for a Variable in a Formula 4. Summary

Formula: A formula is an equation that uses letters to express a relationship between two or more variables.  Example: The formula for finding the perimeter of a rectangle P = 2 l + 2 w where P stands for perimeter, l stands for the length of the rectangle, w stands for the width of the rectangle. This formula expresses the relationship between the length and the width of a rectangle. Common Formulas for Area (A), Perimeter (P), and Volume (V): 1. Formulas Square A = s 2 P = 4s s = side Rectangle A = lw P = 2l + 2w Circle A =  r 2 C = 2  r C = Circumference; r = radius Triangle A = ½ bh b =base; h = height Trapezoid A = ½ h(a + b) Cube V = s 3 S = side Rectangular Solid V = lwh Circular Cylinder V =  r 2 h Sphere V = 4 / 3  r 3 Cone V = 1 / 3  r 2 h s s l w r h b h a b s s s l w h h r r h r

2. Problem Solving With Linear Equations Let us solve a example based on linear equations: Example: The length of the rectangular tennis court at Wimbledon is 6 feet longer than twice the width. If the court’s perimeter is 228 feet, what are the court’s dimensions? Solution: We are given that the length of the rectangular tennis court is 6 feet longer than twice the width and the perimeter of the court is 228 feet. We have to find the length and the breadth of the tennis court. Let us assume that the width of the court is x feet. Now, let us represent the other unknown quantity, the length of the court, in terms of x. The length of the court is 6 feet longer than twice the width (Twice the width = 2 x). Therefore, the length of the court = 2 x + 6 Now we know that the perimeter of a rectangle is given by P = 2l + 2w, where P stands for perimeter, l stands for length and w stands for the width of the rectangle. Next, let us substitute the values of length and width found above in the formula of the perimeter of the rectangle. 228 = 2. (2x + 6) + 2 (x)

2. Problem Solving With Linear Equations (Cont…) Now we solve the equation for x: 228 = 2. (2x + 6) + 2 (x) (Given equation) 2. 2x x = 228 (Using the distributive property) 4x x = 228 (Simplifying the left side of the equation) 6x + 12 = 228 6x + 12 – 12 = (Subtracting 12 from both sides) 6x = 216 x = 36(Dividing both sides by 6) Thus, width of the court is 36 feet. Now let us find the length of the court. The length of the court = (2 x + 6) feet = ( ) feet(Substituting the value of x) = (72 + 6) feet (Simplifying) = 78 feet Thus, the dimensions of the court are 78 feet by 36 feet.

2. Problem Solving With Linear Equations (Cont…) Here are the steps used to solve the previous example: Strategy for problem solving:  Step 1: Read the problem carefully. Attempt to state the problem in your own words. Make a note of what is given in the statement and what has to be determined. Let x (or any variable) denote one of the unknown quantities.  Step 2: If necessary, write expressions for any other unknown quantities in the problem, in terms of x.  Step 3: Write an equation that describes the condition stated in the problem in terms of x.  Step 4: Solve the equation for x.  Step 5: Check the proposed solution in the original wording of the problem, and not in the equation obtained from the words. Let us solve some problems to illustrate these steps: Example 1: Seven subtracted from five times a number is 123. Find the number.  Step 1: Represent one of the unknown quantities by x. Given: Seven subtracted from five times a number = 123. Let the given number be x.  Step 2: Represent other quantities in terms of x. There are no other quantities, so we can skip this step.

2. Problem Solving With Linear Equations (Cont…)  Step 3: Write an equation in x that describes the given conditions. The given condition states that seven subtracted from five times a number is 123. Five times the given number = 5x Seven subtracted from five times the given number = 5x - 7 = 123  Step 4: Solve the equation for x. 5x = (Add 7 to both sides to isolate the variable) 5x = 130(Simplify) x = 130/5(Divide both sides by 5) x = 26 Thus, the number is 26.  Step 5: Check the proposed solution in the original wording of the problem: The original wording of the problem states that seven subtracted from five times a number is 123. The given number = 26 Five times the given number = = 130 Seven subtracted from five times a number = 130 – 7 = 123. Thus, the solution set for the given problem is {26}.

Example 2: The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions?  Step 1: Represent one of the unknown quantities by x. Given: The length of the basketball court is 44 feet more than the width of the court. Also, the perimeter of the court is 288 feet. To be found: The length and the width of the court. Assume the width of the court = x  Step 2: Represent other quantities in terms of x. The other unknown quantity is the length of the court. As the length of the court is 44 feet more than the width of the court, so, the length = x + 44  Step 3: Write an equation in x that describes the given conditions: Here, it is given that the perimeter of the court = 288 feet. Substituting the values in the formula for perimeter of a rectangle, we get 288 = 2. (x + 44) + 2. x Recall: Perimeter of a rectangle = 2l + 2w 2. Problem Solving With Linear Equations (Cont…) Width = x Length = x + 44 Twice the length of the court Twice the width of the court Perimeter of the court

2. Problem Solving With Linear Equations (Cont…)  Step 4: Solve the equation: 2. (x + 44) + 2. x = 288 (Equation that represents the given condition) 2x x = 288 (Using distributive property) 4x + 88 = 288 (Combining like terms) 4x + 88 – 88 = 288 – 88 (Subtracting 88 from both sides) 4x = 200 (Simplifying both sides of the equation) x = 200 / 4 (Dividing both sides by 4) x = 50 Thus, width = 50 feet Length = x + 44 = = 94 feet  Step 5: Check the proposed solution in the original wording of the problem: Let us find out the perimeter of the basketball court using the dimensions Width of the court = 50 feet and Length of the court = 94 feet Perimeter = 2.(94 feet) + 2. (50 feet)(Perimeter = 2.l + 2. w) = 188 feet feet(Simplifying both sides of the equation) = 288 feet As we get the same answer as stated in the problem’s original wording, so our proposed solution is correct.

3. Solving for a Specified Variable in a Formula At times we are required to solve a given formula for a specified variable. We solve a given formula for a particular variable by following these steps:  Step 1: Simplify the given algebraic expression, if necessary.  Step 2: Isolate all the terms with the specified variable on one side of the equation and the terms without the specified variable on the other side of the equation.  Step 3: Solve the given formula for the specified variable. Example: Solve the formula  Step 1: Simplify the given algebraic expression, if necessary.  Step 2: Isolate all the terms with the specified variable on one side.  Step 3: Solve the given formula for the specified variable.

3. Solving for a Specified Variable in a Formula (Cont…) Example 2: Solve IR + Ir = E for I  Step 1: Simplify the given algebraic expression, if necessary. I. R + I. r = E (Given formula) The algebraic expression is already in its simplified form.  Step 2: Isolate all the terms with the specified variable on one side. As all the terms containing the specified variable I are on one side of the equation, this step need not be done for this example.  Step 3: Solve the given formula for the specified variable. We have to solve the given equation for I. I (R + r) = E (Taking I common from both the terms)

Summary Let us recall what we have learned so far: Formula: A formula is an equation that uses letters to express a relationship between two or more variables. Strategy for problem solving using linear equations:  Step 1: Read the problem carefully. Attempt to state the problem in your own words, note what is given in the statement and what is to be determined. Let x (or any variable) denote one of the unknown quantities.  Step 2: If necessary, write expressions for any other unknown quantities in the problem in terms of x.  Step 3: Write an equation in x that describes the conditions.  Step 4: Solve the equation for x.  Step 5: Check the proposed solution in the original wording of the problem, and not in the equation obtained from the words. Steps for solving for a specified variable in a formula:  Step 1: Simplify the given algebraic expression, if necessary.  Step 2: Isolate all the terms with the specified variable on one side of the equation and and the terms without the specified variable on the other side of the equation.  Step 3: Solve for the specified variable.