Rearrange Formulas

lessonbunny.com

Answer Key & Explanations

Rearrange Formulas · Grade 9 · Worksheet 3

1. Rearrange V = (1/3)πr²h to solve for h Answer: h = 3V/(πr²) Solution: Start with the original 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) The rearranged formula is h = 3V/(πr²)
   Full step-by-step solution

   Step 1: Start with the original 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 rearranged formula is h = 3V/(πr²)

2. Kaia is designing a parabolic arch for a sculpture. The arch follows the equation y = -3x^2 + 27, where y is the height in meters and x is the horizontal distance from the center. Rearrange this equation to solve for x in terms of y, and then determine the horizontal distance from the center where the height of the arch is 15 meters. Answer: 2 Solution: Start with the given equation: y = -3x^2 + 27 Subtract 27 from both sides: y - 27 = -3x^2 Divide both sides by -3: (y - 27)/(-3) = x^2 Simplify: x^2 = (27 - y)/3 Take the square root of both sides: x = sqrt((27 - y)/3) or x = -sqrt((27 - y)/3) Since x represents horizontal distance from center,…
   Full step-by-step solution

   Step 1: Start with the given equation: y = -3x^2 + 27 Step 2: Subtract 27 from both sides: y - 27 = -3x^2 Step 3: Divide both sides by -3: (y - 27)/(-3) = x^2 Step 4: Simplify: x^2 = (27 - y)/3 Step 5: Take the square root of both sides: x = sqrt((27 - y)/3) or x = -sqrt((27 - y)/3) Step 6: Since x represents horizontal distance from center, we consider the positive root: x = sqrt((27 - y)/3) Step 7: Substitute y = 15: x = sqrt((27 - 15)/3) Step 8: Simplify inside: x = sqrt(12/3) Step 9: Simplify further: x = sqrt(4) Step 10: x = 2 The horizontal distance from the center where the height is 15 meters is 2 meters.

3. Rearrange P = 2(l + 7) to solve for l Answer: l = (P - 14)/2 Solution: Start with the original formula: P = 2(l + 7) Distribute the 2: P = 2l + 14 Subtract 14 from both sides: P - 14 = 2l Divide both sides by 2: (P - 14)/2 = l Write the final answer: l = (P - 14)/2
   Full step-by-step solution

   Step 1: Start with the original formula: P = 2(l + 7) Step 2: Distribute the 2: P = 2l + 14 Step 3: Subtract 14 from both sides: P - 14 = 2l Step 4: Divide both sides by 2: (P - 14)/2 = l Step 5: Write the final answer: l = (P - 14)/2

4. A cylindrical water tank has a radius of 5 meters and a height of 10 meters. Water is poured into the tank at a constant rate, and the volume of water in the tank is given by V = πr²h, where h is the depth of the water. If the volume of water is 125π cubic meters, what is the depth h of the water in the tank? Answer: 5 Solution: Write down the formula for the volume of a cylinder: V = πr²h. Substitute the given values: V = 125π cubic meters and r = 5 meters. This gives the equation: 125π = π * (5)² * h.
   Full step-by-step solution

   Step 1: Write down the formula for the volume of a cylinder: V = πr²h. Step 2: Substitute the given values: V = 125π cubic meters and r = 5 meters. Step 3: This gives the equation: 125π = π * (5)² * h. Step 4: Simplify the right side: (5)² = 25, so the equation becomes 125π = π * 25 * h, or 125π = 25πh. Step 5: Divide both sides by 25π to isolate h: (125π) / (25π) = h. Step 6: Simplify the left side: 125π divided by 25π equals 125/25 = 5. Step 7: Therefore, h = 5 meters. The answer is 5.

5. A drone is flying at a constant altitude of 120 meters. The drone operator measures the angle of depression to a landing pad as 30°. Using the formula h = d × tan(θ), where h is the altitude, d is the horizontal distance to the landing pad, and θ is the angle of depression, rearrange the formula to solve for d in terms of h and θ. Then calculate the horizontal distance to the landing pad. Answer: d = h / tan(θ); 207.84 meters Solution: In trigonometry problems involving angles of elevation or depression, we often need to rearrange formulas to solve for different variables.
   Full step-by-step solution

   In trigonometry problems involving angles of elevation or depression, we often need to rearrange formulas to solve for different variables. The tangent function relates the opposite side to the adjacent side in a right triangle. When the angle of depression is given, it creates congruent angles between the line of sight and the horizontal, allowing us to use trigonometric ratios in the right triangle formed by the altitude, horizontal distance, and line of sight.

6. 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)