Formula Rearrangement

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Answer Key & Explanations

Formula Rearrangement · Grade 9 · Worksheet 1

1. Solve for x: 7(x - 9)² - 112 = 0 Answer: x = 13 or x = 5 Solution: Add 112 to both sides: 7(x - 9)² = 112 Divide both sides by 7: (x - 9)² = 16 Take the square root of both sides: x - 9 = ±4 Solve for the positive case: x - 9 = 4 → x = 13 Solve for the negative case: x - 9 = -4 → x = 5 The solutions are x = 13 or x = 5.
   Full step-by-step solution

   Step 1: Add 112 to both sides: 7(x - 9)² = 112 Step 2: Divide both sides by 7: (x - 9)² = 16 Step 3: Take the square root of both sides: x - 9 = ±4 Step 4: Solve for the positive case: x - 9 = 4 → x = 13 Step 5: Solve for the negative case: x - 9 = -4 → x = 5 The solutions are x = 13 or x = 5.

2. Liam is designing a rectangular garden with a perimeter of 48 meters. He wants the length to be 6 meters more than twice the width. Write an equation to find the width of the garden, then solve for the width. Answer: 6 Solution: Let the width of the garden be \( w \) meters. Let the length of the garden be \( l \) meters. 1.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. Let the length of the garden be \( l \) meters. --- **Step 2: Translate the problem into equations** We are told: 1. The perimeter is 48 meters. Perimeter formula for a rectangle: \( 2l + 2w = 48 \) 2. The length is 6 meters more than twice the width: \( l = 2w + 6 \) --- **Step 3: Substitute the expression for \( l \) into the perimeter equation** From \( l = 2w + 6 \), substitute into \( 2l + 2w = 48 \): \[ 2(2w + 6) + 2w = 48 \] --- **Step 4: Simplify and solve for \( w \)** First, distribute the 2: \[ 4w + 12 + 2w = 48 \] Combine like terms: \[ 6w + 12 = 48 \] Subtract 12 from both sides: \[ 6w = 36 \] Divide both sides by 6: \[ w = 6 \] --- **Step 5: Interpret the result** The width is 6 meters. We can check: Length \( l = 2(6) + 6 = 18 \) meters. Perimeter = \( 2(18) + 2(6) = 36 + 12 = 48 \) meters. ✓ --- **Final answer:** Width = 6 meters.

3. Aroha is designing a rectangular prism-shaped fish tank with a volume of 1728 cubic inches. The length of the tank is 16 inches, and the width is 9 inches. She needs to find the height of the tank. Write the formula for the volume of a rectangular prism, rearrange it to solve for height, and then calculate the height. Answer: 12 Solution: Write the formula for the volume of a rectangular prism: V = l * w * h. We are given V = 1728 cubic inches, l = 16 inches, w = 9 inches, and we need to solve for h. Rearrange the formula to isolate h.
   Full step-by-step solution

   Step 1: Write the formula for the volume of a rectangular prism: V = l * w * h. Step 2: We are given V = 1728 cubic inches, l = 16 inches, w = 9 inches, and we need to solve for h. Step 3: Rearrange the formula to isolate h. Since V = l * w * h, divide both sides by (l * w): h = V / (l * w). Step 4: Substitute the given values: h = 1728 / (16 * 9). Step 5: Calculate the denominator: 16 * 9 = 144. Step 6: Divide: 1728 / 144 = 12. Step 7: Therefore, the height of the fish tank is 12 inches. The answer is 12.

4. Aroha is designing a rectangular solar panel for a school project. The panel's length is 9 centimeters more than three times its width. The area of the panel is 210 square centimeters. Write an equation in terms of the width w that represents this situation, then solve for the width of Aroha's solar panel. Answer: 7 Solution: The area of a rectangle is A = length * width. Let w = width of the solar panel. The length is 9 cm more than three times the width, so length = 3w + 9.
   Full step-by-step solution

   Step 1: The area of a rectangle is A = length * width. Step 2: Let w = width of the solar panel. Step 3: The length is 9 cm more than three times the width, so length = 3w + 9. Step 4: Substitute into the area formula: 210 = (3w + 9) * w. Step 5: Expand the right side: 210 = 3w^2 + 9w. Step 6: Rearrange to standard quadratic form: 3w^2 + 9w - 210 = 0. Step 7: Divide all terms by 3: w^2 + 3w - 70 = 0. Step 8: Factor the quadratic: (w + 10)(w - 7) = 0. Step 9: Solve for w: w = -10 or w = 7. Step 10: Since width cannot be negative, w = 7. The width of Aroha's solar panel is 7 centimeters.

5. Matiu is building a rectangular fence for his vegetable garden. The area of the garden is 112 square meters. The length is 6 meters longer than the width. Write an equation in terms of the width w that represents this situation, then solve for the width of Matiu's garden. Answer: 8 Solution: The area of a rectangle is A = length × width. Let w = width of the garden in meters. The length is 6 meters longer than the width, so length = w + 6.
   Full step-by-step solution

   Step 1: The area of a rectangle is A = length × width. Step 2: Let w = width of the garden in meters. Step 3: The length is 6 meters longer than the width, so length = w + 6. Step 4: Substitute into the area formula: 112 = (w + 6) × w Step 5: Expand the right side: 112 = w² + 6w Step 6: Rearrange to standard quadratic form: w² + 6w - 112 = 0 Step 7: Factor the quadratic: (w + 14)(w - 8) = 0 Step 8: Solve for w: w + 14 = 0 gives w = -14, or w - 8 = 0 gives w = 8 Step 9: Since width cannot be negative, w = 8 The width of Matiu's garden is 8 meters.

6. Solve for h: V = (1/3)πr²h Answer: h = 3V/(πr²) Solution: Start with V = (1/3)πr²h Multiply both sides by 3 to eliminate the fraction: 3V = πr²h Divide both sides by πr² to isolate h: h = 3V/(πr²) The answer is h = 3V/(πr²).
   Full step-by-step solution

   Step 1: Start with V = (1/3)πr²h Step 2: Multiply both sides by 3 to eliminate the fraction: 3V = πr²h Step 3: Divide both sides by πr² to isolate h: h = 3V/(πr²) The answer is h = 3V/(πr²).

7. Solve for a: S = (1/2)at² + vt Answer: a = 2(S - vt)/t² Solution: Start with the formula S = (1/2)at² + vt Subtract vt from both sides to isolate the term with a: S - vt = (1/2)at² Multiply both sides by 2 to eliminate the fraction: 2(S - vt) = at² Divide both sides by t² to solve for a: a = 2(S - vt)/t² The answer is a = 2(S - vt)/t².
   Full step-by-step solution

   Step 1: Start with the formula S = (1/2)at² + vt Step 2: Subtract vt from both sides to isolate the term with a: S - vt = (1/2)at² Step 3: Multiply both sides by 2 to eliminate the fraction: 2(S - vt) = at² Step 4: Divide both sides by t² to solve for a: a = 2(S - vt)/t² The answer is a = 2(S - vt)/t².

8. Emma is designing a rectangular solar panel for a school project. The length of the panel is 5 meters more than three times its width. The area of the panel is 42 square meters. Write an equation in terms of the width w that represents this situation, then solve for the width of Emma's solar panel. Answer: 3 Solution: Let w = width of the solar panel in meters. The length is 5 meters more than three times the width, so length = 3w + 5. The area of a rectangle is A = length * width.
   Full step-by-step solution

   Step 1: Let w = width of the solar panel in meters. Step 2: The length is 5 meters more than three times the width, so length = 3w + 5. Step 3: The area of a rectangle is A = length * width. Substitute the given area (42) and the expression for length: 42 = (3w + 5) * w. Step 4: Expand the right side: 42 = 3w^2 + 5w. Step 5: Rearrange to standard quadratic form by subtracting 42 from both sides: 3w^2 + 5w - 42 = 0. Step 6: Factor the quadratic. Look for two numbers that multiply to 3 * (-42) = -126 and add to 5. These numbers are 14 and -9. Rewrite the middle term: 3w^2 + 14w - 9w - 42 = 0. Step 7: Group and factor: w(3w + 14) - 3(3w + 14) = 0, so (3w + 14)(w - 3) = 0. Step 8: Set each factor to zero: 3w + 14 = 0 or w - 3 = 0. Solve: w = -14/3 or w = 3. Step 9: Width cannot be negative, so w = 3. The width of Emma's solar panel is 3 meters.