Rearrange Formulas

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Answer Key & Explanations

Rearrange Formulas · Grade 9 · Worksheet 1

1. Rearrange S = 4πr² + 12πrh to solve for h. Answer: h = (S - 4πr²) / (12πr) Solution: Start with the original formula: S = 4πr² + 12πrh Subtract 4πr² from both sides to isolate the term with h: S - 4πr² = 12πrh Divide both sides by 12πr to solve for h: (S - 4πr²) / (12πr) = h Write the final answer: h = (S - 4πr²) / (12πr)
   Full step-by-step solution

   Step 1: Start with the original formula: S = 4πr² + 12πrh Step 2: Subtract 4πr² from both sides to isolate the term with h: S - 4πr² = 12πrh Step 3: Divide both sides by 12πr to solve for h: (S - 4πr²) / (12πr) = h Step 4: Write the final answer: h = (S - 4πr²) / (12πr)

2. Liam is designing a rectangular garden with a fixed perimeter of 40 meters. He wants to express the area A of the garden in terms of its width w. Given that the length l can be written as l = 20 - w, what is the quadratic function A(w) that represents the area of the garden? Answer: A(w) = 20w - w² Solution: Recall the formula for the area of a rectangle. A = length × width Identify the given expressions. Width = w Length l = 20 - w Perimeter is fixed at 40 m, but we already have l in terms of w.
   Full step-by-step solution

   Step 1: Recall the formula for the area of a rectangle. The area A of a rectangle is given by: A = length × width Step 2: Identify the given expressions. Width = w Length l = 20 - w Perimeter is fixed at 40 m, but we already have l in terms of w. Step 3: Substitute the expressions for length and width into the area formula. A = l × w A = (20 - w) × w Step 4: Expand the expression. A = 20 × w - w × w A = 20w - w² Step 5: Write the final quadratic function. A(w) = 20w - w² This is the area of the garden in terms of the width w.

3. 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)

4. Solve for x: 2^(x+1) = 32 Answer: 4 Solution: Write 32 as a power of 2: 32 = 2^5 Substitute into the equation: 2^(x+1) = 2^5 Since the bases are equal, set the exponents equal: x + 1 = 5 Subtract 1 from both sides: x = 5 - 1 x = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Write 32 as a power of 2: 32 = 2^5 Step 2: Substitute into the equation: 2^(x+1) = 2^5 Step 3: Since the bases are equal, set the exponents equal: x + 1 = 5 Step 4: Subtract 1 from both sides: x = 5 - 1 Step 5: x = 4 The answer is 4.

5. A rectangular garden has a length that is 3 meters more than twice its width. If the area of the garden is 65 square meters, find the width of the garden in meters. Answer: 5 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). Area of rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of rectangle = length × width. Given area = 65 m², so: \[ l \times w = 65 \] \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** Use the quadratic formula: For \( a w^2 + b w + c = 0 \), \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here \( a = 2 \), \( b = 3 \), \( c = -65 \). \[ \Delta = b^2 - 4ac = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{\Delta} = \sqrt{529} = 23 \] \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** First: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Second: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) Width cannot be negative, so \( w = 5 \). --- **Step 6: Check** Width \( w = 5 \) m Length \( l = 2(5) + 3 = 13 \) m Area = \( 5 \times 13 = 65 \) m² — correct. --- **Final answer:** 5

6. Aisha is designing a suspension bridge where the cable follows a parabolic path. The cable is anchored at two points 120 meters apart, and the lowest point of the cable is 20 meters below the anchor points. If the vertex of the parabola is at the lowest point, and the origin is placed at the left anchor point, what is the equation of the parabola in vertex form? Answer: y = (1/180)x^2 - (2/3)x Solution: The vertex form of a parabola is y = a(x - h)^2 + k, where (h,k) is the vertex. When given the vertex and another point on the parabola, you can substitute these values to solve for the coefficient 'a'.
   Full step-by-step solution

   The vertex form of a parabola is y = a(x - h)^2 + k, where (h,k) is the vertex. When given the vertex and another point on the parabola, you can substitute these values to solve for the coefficient 'a'. This method works for any parabolic shape, not just bridge cables.

7. Aroha is designing a decorative pendant in the shape of a right circular cone. The pendant has a height of 27 cm and a base radius of 12 cm. A horizontal slice is made through the cone, parallel to the base, at a distance of 9 cm from the vertex. What is the radius of the circular cross-section created by this slice? Answer: 4 Solution: Visualize the problem. The original cone has a height H = 27 cm and a base radius R = 12 cm. The slice is made 9 cm from the vertex.
   Full step-by-step solution

   Step 1: Visualize the problem. The original cone has a height H = 27 cm and a base radius R = 12 cm. The slice is made 9 cm from the vertex. This creates a smaller, similar cone at the top with height h = 9 cm and an unknown radius r. Step 2: Because the slice is parallel to the base, the original cone and the smaller cone at the top are similar. This means the ratios of their corresponding linear dimensions are equal. Step 3: Set up the proportion relating the radii and heights of the two similar cones: (radius of small cone) / (radius of large cone) = (height of small cone) / (height of large cone) r / 12 = 9 / 27 Step 4: Simplify the fraction on the right side: 9 / 27 = 1 / 3 So the equation becomes: r / 12 = 1 / 3 Step 5: Solve for r by multiplying both sides by 12: r = 12 * (1 / 3) r = 12 / 3 r = 4 Step 6: State the answer with units. The radius of the circular cross-section is 4 cm. The answer is 4.