Rearrange Formulas

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Rearrange Formulas · Grade 9 · Worksheet 3

1. Olivia is designing a rectangular garden. The area A of the garden is given by the formula A = l * w, where l is the length and w is the width. She wants to rearrange this formula to solve for the width w in terms of the area A and length l. What is the expression for w? Answer: w = A / l Solution: Start with the given formula: A = l * w To isolate w, divide both sides of the equation by l (assuming l is not zero): A / l = (l * w) / l Simplify the right side: (l * w) / l = w Therefore, w = A / l The answer is w = A / l.
   Full step-by-step solution

   Step 1: Start with the given formula: A = l * w Step 2: To isolate w, divide both sides of the equation by l (assuming l is not zero): A / l = (l * w) / l Step 3: Simplify the right side: (l * w) / l = w Step 4: Therefore, w = A / l The answer is w = A / l.

2. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden is 65 square meters. Write an equation in terms of the width w that represents this situation, then solve for the width of the garden. Answer: 5 Solution: Let \( w \) = width of the garden (in meters). The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define the variables** Let \( w \) = width of the garden (in meters). The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Area = \( l \times w = (2w + 3) \times w \). Given area = 65 square meters: \[ w(2w + 3) = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] Subtract 65 from both sides: \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2, b = 3, c = -65 \). First, discriminant: \[ b^2 - 4ac = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{529} = 23 \] So: \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Case 2: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) --- **Step 6: Interpret the solutions** Width cannot be negative, so we discard \( w = -6.5 \). Thus, width \( w = 5 \) meters. --- **Final check:** If width \( w = 5 \), length \( l = 2(5) + 3 = 13 \). Area = \( 5 \times 13 = 65 \) — correct. --- **Final answer:** 5

3. A rectangular prism has a volume V = 96 cubic units. The prism's height is h units, and its base is a square with side length s units. The volume formula is V = s²h. If the height of the prism is 6 units, what is the side length s of the square base? Answer: 4 Solution: Write down the given formula: V = s²h. Step 2: Substitute the known values: 96 = s² * 6. Step 3: Divide both sides by 6 to isolate s²: 96 / 6 = s² => 16 = s².
   Full step-by-step solution

   Step 1: Write down the given formula: V = s²h. Step 2: Substitute the known values: 96 = s² * 6. Step 3: Divide both sides by 6 to isolate s²: 96 / 6 = s² => 16 = s². Step 4: Take the square root of both sides to solve for s: s = sqrt(16) = 4. Step 5: Since side length is positive, s = 4 units. The answer is 4.

4. Emma is analyzing a rectangular prism with a square base. The volume V of the prism is given by the formula V = s^2 h, where s is the side length of the square base and h is the height. If the volume is 500 cubic centimeters and the height is 20 centimeters, what is the side length s of the square base? Answer: 5 Solution: Start with the formula V = s^2 h. We are given V = 500 and h = 20. We need to solve for s.
   Full step-by-step solution

   Step 1: Start with the formula V = s^2 h. We are given V = 500 and h = 20. We need to solve for s. Step 2: Divide both sides by h to isolate s^2: V/h = s^2, so 500/20 = s^2. Step 3: Simplify the left side: 500/20 = 25, so s^2 = 25. Step 4: Take the square root of both sides to solve for s: s = sqrt(25) = 5. The side length of the square base is 5 centimeters.

5. Rearrange S = 6πr² + 8πrh to solve for h. Answer: h = (S - 6πr²) / (8πr) Solution: Start with the original formula: S = 6πr² + 8πrh Subtract 6πr² from both sides to isolate the term with h: S - 6πr² = 8πrh Divide both sides by 8πr to solve for h: (S - 6πr²) / (8πr) = h Write the final answer: h = (S - 6πr²) / (8πr)
   Full step-by-step solution

   Step 1: Start with the original formula: S = 6πr² + 8πrh Step 2: Subtract 6πr² from both sides to isolate the term with h: S - 6πr² = 8πrh Step 3: Divide both sides by 8πr to solve for h: (S - 6πr²) / (8πr) = h Step 4: Write the final answer: h = (S - 6πr²) / (8πr)

6. A right circular cone has a height of 12 cm and a base radius of 5 cm. A horizontal cross-section is taken at a height of 8 cm from the vertex. What is the radius of this circular cross-section? Answer: 3.33 Solution: - Height \( H = 12 \) cm - Base radius \( R = 5 \) cm - A horizontal cross-section is taken at height \( h = 8 \) cm from the vertex (so from the tip, not from the base). We need the radius \( r \) of that cross-section.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We have a right circular cone with: - Height \( H = 12 \) cm - Base radius \( R = 5 \) cm - A horizontal cross-section is taken at height \( h = 8 \) cm from the **vertex** (so from the tip, not from the base). We need the radius \( r \) of that cross-section. --- **Step 2: Visualize the cone and cross-section** The cone's vertex is at the top (height 0 from vertex), base is at height 12 cm from vertex. At the base: - Height from vertex = 12 cm - Radius = 5 cm At the cross-section: - Height from vertex = 8 cm - Radius = unknown \( r \) --- **Step 3: Use similar triangles** A vertical cross-section through the cone’s axis gives a triangle of height 12 cm and base width 10 cm (since base radius is 5 cm). So half of that triangle (right triangle from axis to side) has: - Height = 12 cm - Horizontal leg = 5 cm At a height \( y \) from the vertex, the radius \( r \) is proportional to the height because the sides are straight lines. --- **Step 4: Set up proportion** From similar triangles: (radius at height y) / (base radius R) = (height from vertex y) / (total height H) So: \( r / 5 = 8 / 12 \) --- **Step 5: Solve for r** \( r / 5 = 8 / 12 \) Simplify \( 8 / 12 = 2 / 3 \) \( r / 5 = 2 / 3 \) \( r = 5 \times (2 / 3) \) \( r = 10 / 3 \) --- **Step 6: Convert to decimal** \( 10 / 3 = 3.333... \) Rounded to two decimal places: \( 3.33 \) --- **Final answer:** 3.33

7. Rearrange S = 2πr² + 12πrh to solve for h. Answer: h = (S - 2πr²)/(12πr) Solution: Start with the original formula: S = 2πr² + 12πrh Subtract 2πr² from both sides to isolate the term with h: S - 2πr² = 12πrh Divide both sides by 12πr to solve for h: (S - 2πr²)/(12πr) = h Write the final answer: h = (S - 2πr²)/(12πr)
   Full step-by-step solution

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