Rearrange Formulas

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Rearrange Formulas · Grade 9 · Worksheet 2

1. Liam is designing a rectangular garden with a fixed perimeter of 40 meters. He wants to express the area A of the garden as a function of its width w. The length of the garden can be determined from the perimeter constraint. Write the formula for A in terms of w. Answer: A = 20w - w^2 Solution: We have a rectangular garden with a fixed perimeter of 40 meters.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We have a rectangular garden with a fixed perimeter of 40 meters. Let: - \( w \) = width (in meters) - \( l \) = length (in meters) The perimeter formula for a rectangle is: \[ P = 2l + 2w \] Given \( P = 40 \), we have: \[ 2l + 2w = 40 \] --- **Step 2: Solve for length \( l \) in terms of width \( w \)** Divide both sides of \( 2l + 2w = 40 \) by 2: \[ l + w = 20 \] Subtract \( w \) from both sides: \[ l = 20 - w \] --- **Step 3: Write the area \( A \) in terms of \( w \)** The area of a rectangle is: \[ A = l \times w \] Substitute \( l = 20 - w \): \[ A = (20 - w) \times w \] --- **Step 4: Simplify the expression** \[ A = 20w - w^2 \] --- **Final Answer:** \[ A = 20w - w^2 \] This is the area as a function of the width \( w \).

2. Aisha is designing a skateboard ramp with a triangular cross-section. The ramp's height is 2 meters less than its base length, and the area of the triangular cross-section is 12 square meters. If Aisha represents the base length as b meters, what equation should she use to find the dimensions of her ramp? Answer: 1/2 * b * (b - 2) = 12 Solution: When working with geometric shapes where dimensions are related, we often use formulas for area or perimeter and substitute the relationships between variables.
   Full step-by-step solution

   When working with geometric shapes where dimensions are related, we often use formulas for area or perimeter and substitute the relationships between variables. For triangles specifically, the area formula connects base and height, allowing us to create equations when one dimension is expressed in terms of another. This approach is useful in many design and engineering applications where proportions matter.

3. √(x² - 10x + 25) = 7 Answer: 12 Solution: Recognize that x² - 10x + 25 is a perfect square trinomial Factor the expression: x² - 10x + 25 = (x - 5)² Rewrite the equation: √((x - 5)²) = 7 Simplify the square root: |x - 5| = 7 Solve the absolute value equation: x - 5 = 7 or x - 5 = -7 Solve the first case: x - 5 = 7 → x = 12 Solve the…
   Full step-by-step solution

   Step 1: Recognize that x² - 10x + 25 is a perfect square trinomial Step 2: Factor the expression: x² - 10x + 25 = (x - 5)² Step 3: Rewrite the equation: √((x - 5)²) = 7 Step 4: Simplify the square root: |x - 5| = 7 Step 5: Solve the absolute value equation: x - 5 = 7 or x - 5 = -7 Step 6: Solve the first case: x - 5 = 7 → x = 12 Step 7: Solve the second case: x - 5 = -7 → x = -2 Step 8: Check both solutions in the original equation Step 9: For x = 12: √(144 - 120 + 25) = √49 = 7 ✓ Step 10: For x = -2: √(4 + 20 + 25) = √49 = 7 ✓ Both solutions are valid, but the problem asks for a single answer. The positive solution is 12.

4. Liam is designing a rectangular garden with a perimeter of 48 meters. He wants the length to be 6 meters more than the width. Write an equation for the width w in terms of the perimeter, then determine the dimensions of the garden. Answer: width = 9 meters, length = 15 meters Solution: - Perimeter of rectangle = 48 meters - Length is 6 meters more than the width - Let width = w - Then length = w + 6 Perimeter P = 2 × (length + width) Substitute P = 48, length = w + 6, width = w: 48 = 2 × ( (w + 6) + w ) Inside the parentheses: (w + 6 + w) = 2w + 6 So: 48 = 2 × (2w + 6) Divide…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We know: - Perimeter of rectangle = 48 meters - Length is 6 meters more than the width - Let width = w - Then length = w + 6 --- **Step 2: Recall the perimeter formula for a rectangle** Perimeter P = 2 × (length + width) Substitute P = 48, length = w + 6, width = w: 48 = 2 × ( (w + 6) + w ) --- **Step 3: Simplify the equation** Inside the parentheses: (w + 6 + w) = 2w + 6 So: 48 = 2 × (2w + 6) --- **Step 4: Solve for w** Divide both sides by 2: 48 / 2 = 2w + 6 24 = 2w + 6 Subtract 6 from both sides: 24 - 6 = 2w 18 = 2w Divide by 2: w = 9 --- **Step 5: Find length** Length = w + 6 = 9 + 6 = 15 --- **Step 6: Check** Perimeter = 2 × (length + width) = 2 × (15 + 9) = 2 × 24 = 48 meters ✔ Length (15) is 6 more than width (9) ✔ --- **Final answer:** width = 9 meters, length = 15 meters

5. The formula for the volume of a cone is V = (1/3)πr²h. A cone has a volume of 48π cubic centimeters and a height of 9 centimeters. Rearrange the formula to solve for the radius r, and then find the radius of this cone. Answer: 4 Solution: V = (1/3) π r² h V = 48π cubic centimeters h = 9 centimeters 48π = (1/3) π r² (9) First, multiply both sides by 3 to eliminate the 1/3: 3 × 48π = π r² (9) 144π = 9π r² 144 = 9 r² Divide both sides by 9 144 / 9 = r² 16 = r² r = √16 r = 4 Final Answer: The radius is 4 centimeters.
   Full step-by-step solution

   We are given the volume formula for a cone: V = (1/3) π r² h Given: V = 48π cubic centimeters h = 9 centimeters --- **Step 1: Substitute the given values into the formula** 48π = (1/3) π r² (9) --- **Step 2: Simplify the equation** First, multiply both sides by 3 to eliminate the 1/3: 3 × 48π = π r² (9) 144π = 9π r² --- **Step 3: Divide both sides by π** 144 = 9 r² --- **Step 4: Divide both sides by 9** 144 / 9 = r² 16 = r² --- **Step 5: Take the square root** r = √16 r = 4 --- **Final Answer:** The radius is 4 centimeters.

6. Rearrange S = 5πr² + 10πrh to solve for h. Answer: h = (S - 5πr²) / (10πr) Solution: Start with the original formula: S = 5πr² + 10πrh Subtract 5πr² from both sides to isolate the term with h: S - 5πr² = 10πrh Divide both sides by 10πr to solve for h: (S - 5πr²) / (10πr) = h Write the final answer: h = (S - 5πr²) / (10πr)
   Full step-by-step solution

   Step 1: Start with the original formula: S = 5πr² + 10πrh Step 2: Subtract 5πr² from both sides to isolate the term with h: S - 5πr² = 10πrh Step 3: Divide both sides by 10πr to solve for h: (S - 5πr²) / (10πr) = h Step 4: Write the final answer: h = (S - 5πr²) / (10πr)

7. Rearrange A = 2πr² + 2πrh to solve for h Answer: (A - 2πr²)/(2πr) Solution: Start with the original formula: A = 2πr² + 2πrh Subtract 2πr² from both sides: A - 2πr² = 2πrh Divide both sides by 2πr: (A - 2πr²)/(2πr) = h The formula rearranged for h is: h = (A - 2πr²)/(2πr)
   Full step-by-step solution

   Step 1: Start with the original formula: A = 2πr² + 2πrh Step 2: Subtract 2πr² from both sides: A - 2πr² = 2πrh Step 3: Divide both sides by 2πr: (A - 2πr²)/(2πr) = h Step 4: The formula rearranged for h is: h = (A - 2πr²)/(2πr)

8. Aisha is designing a triangular garden with a base that is 2 meters longer than its height. The area of the garden is 24 square meters. Write an equation in terms of the height h that represents this situation, then solve for the height of the garden in meters. Answer: 6 Solution: Recall the area formula for a triangle: Area = (1/2) × base × height The base is 2 meters longer than the height, so base = h + 2 Substitute into the area formula: 24 = (1/2) × (h + 2) × h Multiply both sides by 2: 48 = (h + 2) × h Expand the right side: 48 = h² + 2h Rearrange into standard…
   Full step-by-step solution

   Step 1: Recall the area formula for a triangle: Area = (1/2) × base × height Step 2: Let h represent the height in meters Step 3: The base is 2 meters longer than the height, so base = h + 2 Step 4: Substitute into the area formula: 24 = (1/2) × (h + 2) × h Step 5: Multiply both sides by 2: 48 = (h + 2) × h Step 6: Expand the right side: 48 = h² + 2h Step 7: Rearrange into standard quadratic form: h² + 2h - 48 = 0 Step 8: Factor the quadratic: (h + 8)(h - 6) = 0 Step 9: Solve for h: h = -8 or h = 6 Step 10: Since height cannot be negative, h = 6 The height of the garden is 6 meters.