Formula Rearrangement

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Formula Rearrangement · Grade 9 · Worksheet 2

1. Aisha is designing a triangular garden with sides that form a right triangle. The hypotenuse is 5 meters longer than the longer leg, and the longer leg is 2 meters longer than the shorter leg. Write an equation in terms of the shorter leg length x that represents this situation using the Pythagorean theorem, then solve for x to find the length of the shorter leg. Answer: 5 Solution: Step 1: Let x = length of shorter leg Step 2: Longer leg = x + 2 Step 3: Hypotenuse = (x + 2) + 5 = x + 7 Step 4: Apply Pythagorean theorem: (shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2 Step 5: x^2 + (x + 2)^2 = (x + 7)^2 Step 6: Expand: x^2 + (x^2 + 4x + 4) = x^2 + 14x + 49 Step 7:…
   Full step-by-step solution

   Step 1: Let x = length of shorter leg Step 2: Longer leg = x + 2 Step 3: Hypotenuse = (x + 2) + 5 = x + 7 Step 4: Apply Pythagorean theorem: (shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2 Step 5: x^2 + (x + 2)^2 = (x + 7)^2 Step 6: Expand: x^2 + (x^2 + 4x + 4) = x^2 + 14x + 49 Step 7: Simplify: 2x^2 + 4x + 4 = x^2 + 14x + 49 Step 8: Subtract right side: x^2 - 10x - 45 = 0 Step 9: Solve using quadratic formula: x = [10 ± sqrt(100 + 180)]/2 = [10 ± sqrt(280)]/2 = [10 ± 2sqrt(70)]/2 = 5 ± sqrt(70) Step 10: Since length must be positive: x = 5 + sqrt(70) ≈ 13.37 or x = 5 - sqrt(70) ≈ -3.37 Step 11: Discard negative solution: x = 5 + sqrt(70) Step 12: Check: shorter leg ≈ 13.37, longer leg ≈ 15.37, hypotenuse ≈ 20.37 Step 13: Verify: 13.37^2 + 15.37^2 = 178.8 + 236.2 = 414.9, 20.37^2 = 414.9 ✓ The length of the shorter leg is 5 + sqrt(70) meters.

2. The formula for the surface area of a cylinder is S = 2πr² + 2πrh. Solve for h. Answer: h = (S - 2πr²) / (2πr) Solution: Start with S = 2πr² + 2πrh. Subtract 2πr² from both sides: S - 2πr² = 2πrh. Divide both sides by 2πr: (S - 2πr²) / (2πr) = h.
   Full step-by-step solution

   Step 1: Start with S = 2πr² + 2πrh. Step 2: Subtract 2πr² from both sides: S - 2πr² = 2πrh. Step 3: Divide both sides by 2πr: (S - 2πr²) / (2πr) = h. Step 4: The solution is h = (S - 2πr²) / (2πr).

3. Tane is designing a rectangular solar panel for a school project. The length of the panel is 9 centimeters more than three times its width. If 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 Tane's solar panel. Answer: 7 Solution: Let w = width of the solar panel in centimeters. The length is 9 centimeters more than three times the width, so length = 3w + 9. Area of a rectangle = length × width = (3w + 9) × w.
   Full step-by-step solution

   Step 1: Let w = width of the solar panel in centimeters. Step 2: The length is 9 centimeters more than three times the width, so length = 3w + 9. Step 3: Area of a rectangle = length × width = (3w + 9) × w. Step 4: Set the area equal to 210: (3w + 9)w = 210. Step 5: Expand the left side: 3w² + 9w = 210. Step 6: Subtract 210 from both sides: 3w² + 9w - 210 = 0. Step 7: Divide the entire equation by 3: w² + 3w - 70 = 0. Step 8: Factor the quadratic: (w + 10)(w - 7) = 0. Step 9: Set each factor to zero: w + 10 = 0 or w - 7 = 0, giving w = -10 or w = 7. Step 10: Width cannot be negative, so w = 7. The width of Tane's solar panel is 7 centimeters.

4. Aroha is building a rectangular skateboard ramp. The area of the ramp's surface is given by the formula A = lw, where l is the length and w is the width. The length of the ramp is 3 times its width. If the area of the ramp is 147 square meters, write an equation in terms of the width w that represents this situation, then solve for the width of Aroha's ramp. Answer: 7 Solution: Let w = width of the ramp in meters. The length l is 3 times the width, so l = 3w. The area formula is A = lw.
   Full step-by-step solution

   Step 1: Let w = width of the ramp in meters. The length l is 3 times the width, so l = 3w. Step 2: The area formula is A = lw. Substitute A = 147 and l = 3w: 147 = (3w) * w Step 3: Simplify the right side: 147 = 3w^2 Step 4: Divide both sides by 3: 147 / 3 = 3w^2 / 3 → 49 = w^2 Step 5: Take the square root of both sides: w = sqrt(49) or w = -sqrt(49) Step 6: Since width cannot be negative, w = 7. The width of Aroha's ramp is 7 meters.

5. Solve for r: A = πr² + 2πrh Answer: r = (-2πh ± √(4π²h² + 4πA)) / (2π) Solution: Start with A = πr² + 2πrh. Rearrange to standard quadratic form: πr² + 2πrh - A = 0. Identify coefficients: a = π, b = 2πh, c = -A.
   Full step-by-step solution

   Step 1: Start with A = πr² + 2πrh. Step 2: Rearrange to standard quadratic form: πr² + 2πrh - A = 0. Step 3: Identify coefficients: a = π, b = 2πh, c = -A. Step 4: Apply the quadratic formula: r = (-b ± √(b² - 4ac)) / (2a). Step 5: Substitute: r = (-2πh ± √((2πh)² - 4π(-A))) / (2π). Step 6: Simplify inside the square root: (2πh)² = 4π²h², and -4π(-A) = +4πA. Step 7: So r = (-2πh ± √(4π²h² + 4πA)) / (2π). Step 8: This is the final expression for r.

6. The formula for the volume of a cone is V = (1/3)πr²h. Solve for h. Answer: h = 3V / (πr²) Solution: Start with the formula 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).
   Full step-by-step solution

   Step 1: Start with the formula 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 r: S = 4πr² Answer: r = sqrt(S / (4π)) Solution: Start with S = 4πr² Divide both sides by 4π to isolate r²: S / (4π) = r² Take the square root of both sides: r = sqrt(S / (4π)) The answer is r = sqrt(S / (4π)).
   Full step-by-step solution

   Step 1: Start with S = 4πr² Step 2: Divide both sides by 4π to isolate r²: S / (4π) = r² Step 3: Take the square root of both sides: r = sqrt(S / (4π)) The answer is r = sqrt(S / (4π)).

8. 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.