Rearrange Formulas

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Rearrange Formulas · Grade 9 · Worksheet 2

1. Liam is analyzing a circular fountain in a park. The fountain has an area of 121π square meters. He needs to determine the radius of the fountain. The formula for the area of a circle is A = πr². Rearrange this formula to solve for the radius r, and then calculate the radius of the fountain. Answer: 11 Solution: Start with the formula for the area of a circle: A = πr². We need to solve for r. First, divide both sides by π to undo the multiplication: A/π = r².
   Full step-by-step solution

   Step 1: Start with the formula for the area of a circle: A = πr². Step 2: We need to solve for r. First, divide both sides by π to undo the multiplication: A/π = r². Step 3: Now, take the square root of both sides to undo the squaring: r = sqrt(A/π). Step 4: Substitute the given area A = 121π into the rearranged formula: r = sqrt(121π / π). Step 5: Simplify inside the square root: 121π / π = 121, so r = sqrt(121). Step 6: Calculate the square root: sqrt(121) = 11. The radius of the fountain is 11 meters.

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. If the length is 5 meters less than twice the width, write a formula for the area A in terms of w only. Answer: A = 2w^2 - 15w Solution: When working with geometric formulas, you can often express one variable in terms of another using given relationships.
   Full step-by-step solution

   When working with geometric formulas, you can often express one variable in terms of another using given relationships. For area problems, start with the basic area formula, then use perimeter information and any additional relationships between dimensions to eliminate variables until you have a single-variable expression.

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

   Step 1: Start with the original formula: S = 2πr² + 7πrh Step 2: Subtract 2πr² from both sides to isolate the term with h: S - 2πr² = 7πrh Step 3: Divide both sides by 7πr to solve for h: (S - 2πr²)/(7πr) = h Step 4: Write the final answer: h = (S - 2πr²)/(7πr)

4. Mason is designing a cylindrical water tank for a community project. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. The tank must have a volume of 1078 cubic feet and a height of 7 feet. Rearrange the formula to solve for the radius r in terms of V, π, and h, then calculate the radius of the tank in feet. Use π = 22/7. Answer: 7 Solution: Start with the formula V = πr²h Divide both sides by πh to isolate r²: r² = V/(πh) Substitute the given values: V = 1078, h = 7, π = 22/7 r² = 1078 / ((22/7) × 7) Simplify the denominator: (22/7) × 7 = 22 r² = 1078 / 22 Divide: 1078 ÷ 22 = 49 r² = 49 Take the square root of both sides: r = √49 =…
   Full step-by-step solution

   Step 1: Start with the formula V = πr²h Step 2: Divide both sides by πh to isolate r²: r² = V/(πh) Step 3: Substitute the given values: V = 1078, h = 7, π = 22/7 Step 4: r² = 1078 / ((22/7) × 7) Step 5: Simplify the denominator: (22/7) × 7 = 22 Step 6: r² = 1078 / 22 Step 7: Divide: 1078 ÷ 22 = 49 Step 8: r² = 49 Step 9: Take the square root of both sides: r = √49 = 7 The radius of the tank is 7 feet.

5. Liam is designing a rectangular garden with an area of 48 square meters. The length of the garden is 4 meters more than its width. Write an equation in terms of width w that represents this situation, then solve for the dimensions of Liam's garden. Answer: width = 6 meters, length = 10 meters Solution: Let \( w \) = width of the garden (in meters). The length is 4 meters more than the width, so: length \( l = w + 4 \). 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 4 meters more than the width, so: length \( l = w + 4 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 48 square meters: \[ w \times (w + 4) = 48 \] --- **Step 3: Expand and rearrange** \[ w^2 + 4w = 48 \] \[ w^2 + 4w - 48 = 0 \] --- **Step 4: Solve the quadratic equation** We solve \( w^2 + 4w - 48 = 0 \) by factoring: Look for two numbers that multiply to -48 and add to 4. Those numbers are 8 and -6. So: \[ (w + 8)(w - 6) = 0 \] --- **Step 5: Find possible values of w** \[ w + 8 = 0 \quad \text{or} \quad w - 6 = 0 \] \[ w = -8 \quad \text{or} \quad w = 6 \] Since width cannot be negative, \( w = 6 \). --- **Step 6: Find the length** \[ l = w + 4 = 6 + 4 = 10 \] --- **Final answer:** Width = 6 meters, Length = 10 meters.

6. Rearrange V = (1/3)πr²h to solve for h. 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).
   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 rearranged formula is h = 3V/(πr²).

7. Sophia is studying the kinetic energy of a moving object. The formula for kinetic energy is KE = 1/2 m v^2, where KE is the kinetic energy in joules, m is the mass in kilograms, and v is the speed in meters per second. Sophia measures a moving object with a kinetic energy of 176 joules and a mass of 6 kilograms. Rearrange the formula to solve for v in terms of KE and m, then calculate the speed of the object. Answer: sqrt(176/3) m/s Solution: Start with the formula KE = 1/2 m v^2. Multiply both sides by 2: 2 KE = m v^2. Divide both sides by m: (2 KE)/m = v^2.
   Full step-by-step solution

   Step 1: Start with the formula KE = 1/2 m v^2. Step 2: Multiply both sides by 2: 2 KE = m v^2. Step 3: Divide both sides by m: (2 KE)/m = v^2. Step 4: Take the square root of both sides: v = sqrt((2 KE)/m). Step 5: Substitute KE = 176 and m = 6: v = sqrt((2 * 176)/6) = sqrt(352/6). Step 6: Simplify the fraction: 352/6 = 176/3. Step 7: v = sqrt(176/3) meters per second. Step 8: The speed is sqrt(176/3) m/s, or approximately 7.66 m/s. Answer: sqrt(176/3) m/s