Formula Rearrangement

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Formula Rearrangement · Grade 9 · Worksheet 3

1. Aisha is designing a triangular support structure for a bridge. The area of the triangular section is 84 square meters, and the base is 7 meters longer than the height. Write an equation in terms of the height h that represents this situation, then solve for the height of the triangular section. Answer: h = 8 Solution: The area of a triangle is given by the formula A = (1/2) * base * height. When the base is expressed in terms of the height, this formula becomes a quadratic equation.
   Full step-by-step solution

   The area of a triangle is given by the formula A = (1/2) * base * height. When the base is expressed in terms of the height, this formula becomes a quadratic equation. Solving such equations often involves using the quadratic formula or factoring, which requires setting the equation to zero and finding the positive root since physical dimensions cannot be negative.

2. Solve for h: V = 2πr² + 2πrh Answer: h = (V - 2πr²) / (2πr) Solution: Start with the formula V = 2πr² + 2πrh. Subtract 2πr² from both sides to isolate the term with h: V - 2πr² = 2πrh. Divide both sides by 2πr to solve for h: h = (V - 2πr²) / (2πr).
   Full step-by-step solution

   Step 1: Start with the formula V = 2πr² + 2πrh. Step 2: Subtract 2πr² from both sides to isolate the term with h: V - 2πr² = 2πrh. Step 3: Divide both sides by 2πr to solve for h: h = (V - 2πr²) / (2πr). The answer is h = (V - 2πr²) / (2πr).

3. The formula for the volume of a sphere is V = (4/3)πr³. If a sphere has a volume of 288π cubic centimeters, what is its radius in centimeters? Answer: 6 Solution: Start with the volume formula: V = (4/3)πr³ Substitute V = 288π: 288π = (4/3)πr³ Divide both sides by π: 288 = (4/3)r³ Multiply both sides by 3/4: 288 × (3/4) = r³ Calculate 288 × (3/4) = 216: 216 = r³ Take the cube root of both sides: r = ∛216 Calculate the cube root: ∛216 = 6 The answer is 6.
   Full step-by-step solution

   Step 1: Start with the volume formula: V = (4/3)πr³ Step 2: Substitute V = 288π: 288π = (4/3)πr³ Step 3: Divide both sides by π: 288 = (4/3)r³ Step 4: Multiply both sides by 3/4: 288 × (3/4) = r³ Step 5: Calculate 288 × (3/4) = 216: 216 = r³ Step 6: Take the cube root of both sides: r = ∛216 Step 7: Calculate the cube root: ∛216 = 6 The answer is 6.

4. Solve for h: V = πr²h Answer: h = V/(πr²) Solution: Start with the formula V = πr²h. To isolate h, divide both sides by πr² (since πr² is multiplied by h). V / (πr²) = (πr²h) / (πr²) Simplify the right side: πr² cancels, leaving h.
   Full step-by-step solution

   Step 1: Start with the formula V = πr²h. Step 2: To isolate h, divide both sides by πr² (since πr² is multiplied by h). Step 3: V / (πr²) = (πr²h) / (πr²) Step 4: Simplify the right side: πr² cancels, leaving h. Step 5: Therefore, h = V / (πr²). The answer is h = V/(πr²).

5. Aisha is designing a triangular garden with a right angle. The hypotenuse is 13 meters long, and one leg is 7 meters longer than the other. Write an equation in terms of the shorter leg length w that represents this situation, then solve for w to find the length of the shorter leg. Answer: 5 Solution: Let w be the length of the shorter leg. The other leg is 7 meters longer, so its length is w + 7. The hypotenuse is 13 meters.
   Full step-by-step solution

   Step 1: Let w be the length of the shorter leg. The other leg is 7 meters longer, so its length is w + 7. The hypotenuse is 13 meters. Step 2: Apply the Pythagorean theorem: (shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2 Step 3: Substitute the expressions: w^2 + (w + 7)^2 = 13^2 Step 4: Expand the equation: w^2 + (w^2 + 14w + 49) = 169 Step 5: Combine like terms: 2w^2 + 14w + 49 = 169 Step 6: Subtract 169 from both sides: 2w^2 + 14w + 49 - 169 = 0 Step 7: Simplify: 2w^2 + 14w - 120 = 0 Step 8: Divide the entire equation by 2: w^2 + 7w - 60 = 0 Step 9: Factor the quadratic: (w + 12)(w - 5) = 0 Step 10: Solve for w: w = -12 or w = 5. Since a length cannot be negative, w = 5. The length of the shorter leg is 5 meters.

6. Liam is designing a rectangular garden with an area of 54 square meters. The length of the garden is 3 meters more than twice its width. Write an equation in terms of the width w that represents this situation, then solve for the width of Liam's garden. Answer: 4.5 Solution: Let’s go step-by-step. Let \( w \) = width of the garden in meters. The length \( l \) is 3 meters more than twice the width, so: \( l = 2w + 3 \).
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Define variables** Let \( w \) = width of the garden in meters. The length \( l \) is 3 meters more than twice the width, so: \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 54 m², so: \( w \times (2w + 3) = 54 \). --- **Step 3: Expand and rearrange** \( w(2w + 3) = 54 \) \( 2w^2 + 3w = 54 \) \( 2w^2 + 3w - 54 = 0 \). --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here \( a = 2 \), \( b = 3 \), \( c = -54 \). First, discriminant: \( b^2 - 4ac = 3^2 - 4(2)(-54) = 9 + 432 = 441 \). \( \sqrt{441} = 21 \). So: \( w = \frac{-3 \pm 21}{2 \times 2} = \frac{-3 \pm 21}{4} \). --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \). Case 2: \( w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \). --- **Step 6: Interpret results** Width cannot be negative, so \( w = 4.5 \) meters. --- **Final answer:** Width = 4.5 meters.

7. Aroha is designing a rectangular sandbox for her community garden. The length of the sandbox is 5 meters more than three times its width. The area of the sandbox is 42 square meters. Write an equation in terms of the width w that represents this situation, then solve for the width of Aroha's sandbox. Answer: 3 Solution: Let w represent the width of the sandbox 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 x width.
   Full step-by-step solution

   Step 1: Let w represent the width of the sandbox 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 x width. Substitute the given area and expressions: 42 = (3w + 5) * w. Step 4: Expand the right side: 42 = 3w^2 + 5w. Step 5: Rearrange into standard quadratic form: 3w^2 + 5w - 42 = 0. Step 6: Factor the quadratic: (3w + 14)(w - 3) = 0. Step 7: Set each factor equal to zero: 3w + 14 = 0 or w - 3 = 0. Step 8: Solve: w = -14/3 or w = 3. Step 9: Since width cannot be negative, the width is 3 meters. The width of Aroha's sandbox is 3 meters.

8. Noah is looking at a rectangular prism with a square base. The volume of the prism is given by the formula V = s²h, where s is the side length of the square base and h is the height. The prism has a volume of 576 cubic units and a height of 16 units. Solve for the side length s of the square base. Answer: 6 Solution: Start with the formula V = s²h. Substitute the given values: 576 = s² * 16. Divide both sides by 16 to isolate s²: s² = 576 / 16 = 36.
   Full step-by-step solution

   Step 1: Start with the formula V = s²h. Step 2: Substitute the given values: 576 = s² * 16. Step 3: Divide both sides by 16 to isolate s²: s² = 576 / 16 = 36. Step 4: Take the square root of both sides to solve for s: s = sqrt(36) = 6. Step 5: Since s is a side length, it is positive, so s = 6 units. The answer is 6.