Formula Rearrangement

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Answer Key & Explanations

Formula Rearrangement · Grade 9 · Worksheet 1

1. Solve for t: d = 4t² + 8t Answer: t = (-2 + sqrt(4 + d)) / 2 or t = (-2 - sqrt(4 + d)) / 2 Solution: Start with the equation d = 4t² + 8t. Rearrange into standard quadratic form: 4t² + 8t - d = 0. Use the quadratic formula t = [-b ± sqrt(b² - 4ac)] / (2a), where a = 4, b = 8, c = -d.
   Full step-by-step solution

   Step 1: Start with the equation d = 4t² + 8t. Step 2: Rearrange into standard quadratic form: 4t² + 8t - d = 0. Step 3: Use the quadratic formula t = [-b ± sqrt(b² - 4ac)] / (2a), where a = 4, b = 8, c = -d. Step 4: Substitute: t = [-8 ± sqrt(8² - 4*4*(-d))] / (2*4) = [-8 ± sqrt(64 + 16d)] / 8. Step 5: Simplify the square root: sqrt(64 + 16d) = sqrt(16(4 + d)) = 4 sqrt(4 + d). Step 6: So t = [-8 ± 4 sqrt(4 + d)] / 8. Step 7: Divide numerator and denominator by 4: t = [-2 ± sqrt(4 + d)] / 2. The solutions are t = (-2 + sqrt(4 + d)) / 2 and t = (-2 - sqrt(4 + d)) / 2.

2. Matiu is a conservation biologist studying a population of endangered birds. The population size, P, after t years can be modeled by the formula P = P0 * e^(kt), where P0 is the initial population and k is the growth rate constant. Matiu observes that an initial population of 15 birds grows to 45 birds in 6 years. Write an equation that represents this situation, and solve for the growth rate constant k, expressing your answer in exact form using natural logarithms. Answer: k = (ln 3) / 6 Solution: Write down the exponential growth formula: P = P0 * e^(kt) Substitute the given values: P = 45, P0 = 15, and t = 6.
   Full step-by-step solution

   Step 1: Write down the exponential growth formula: P = P0 * e^(kt) Step 2: Substitute the given values: P = 45, P0 = 15, and t = 6. This gives: 45 = 15 * e^(6k) Step 3: Divide both sides by 15 to isolate the exponential term: 45 / 15 = e^(6k) 3 = e^(6k) Step 4: Apply the natural logarithm (ln) to both sides of the equation to solve for the exponent: ln(3) = ln(e^(6k)) Step 5: Simplify the right side using the property ln(e^x) = x: ln(3) = 6k Step 6: Divide both sides by 6 to solve for k: k = ln(3) / 6 The exact value of the growth rate constant k is ln(3) / 6.

3. Solve for x: 2(x - 4)² - 72 = 0 Answer: 10 or -2 Solution: Add 72 to both sides: 2(x - 4)² = 72 Divide both sides by 2: (x - 4)² = 36 Take the square root of both sides: x - 4 = ±6 Solve for x: x = 4 ± 6 For the positive case: x = 4 + 6 = 10 For the negative case: x = 4 - 6 = -2 The solutions are x = 10 or x = -2.
   Full step-by-step solution

   Step 1: Add 72 to both sides: 2(x - 4)² = 72 Step 2: Divide both sides by 2: (x - 4)² = 36 Step 3: Take the square root of both sides: x - 4 = ±6 Step 4: Solve for x: x = 4 ± 6 Step 5: For the positive case: x = 4 + 6 = 10 Step 6: For the negative case: x = 4 - 6 = -2 The solutions are x = 10 or x = -2.

4. 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.

5. Solve for x: 4(x - 2)² - 64 = 0 Answer: 6 Solution: Add 64 to both sides: 4(x - 2)² = 64 Divide both sides by 4: (x - 2)² = 16 Take square root of both sides: x - 2 = ±4 Solve for x: x = 2 + 4 OR x = 2 - 4 Calculate both solutions: x = 6 OR x = -2 The positive solution is x = 6.
   Full step-by-step solution

   Step 1: Add 64 to both sides: 4(x - 2)² = 64 Step 2: Divide both sides by 4: (x - 2)² = 16 Step 3: Take square root of both sides: x - 2 = ±4 Step 4: Solve for x: x = 2 + 4 OR x = 2 - 4 Step 5: Calculate both solutions: x = 6 OR x = -2 The positive solution is x = 6.

6. The formula for the volume of a rectangular prism is V = lwh. Solve for w. Answer: w = V/(lh) Solution: Start with the formula V = lwh. To solve for w, divide both sides of the equation by lh (since w is multiplied by l and h). V/(lh) = (lwh)/(lh) Simplify the right side: (lwh)/(lh) = w.
   Full step-by-step solution

   Step 1: Start with the formula V = lwh. Step 2: To solve for w, divide both sides of the equation by lh (since w is multiplied by l and h). Step 3: V/(lh) = (lwh)/(lh) Step 4: Simplify the right side: (lwh)/(lh) = w. Step 5: Therefore, w = V/(lh). The answer is w = V/(lh).

7. Liam is designing a rectangular garden with a perimeter of 40 meters. The length of the garden is 4 meters more than twice its width. Write an equation for the perimeter in terms of the width w, then solve for w to find the garden's dimensions. Answer: w = 6 meters, length = 16 meters Solution: When solving for dimensions given a perimeter and a relationship between length and width, you can use algebraic substitution. The perimeter formula for a rectangle is a fundamental relationship.
   Full step-by-step solution

   When solving for dimensions given a perimeter and a relationship between length and width, you can use algebraic substitution. The perimeter formula for a rectangle is a fundamental relationship. By expressing one dimension in terms of the other based on the given relationship, you create a single equation that can be solved for the unknown variable, then used to find both dimensions.

8. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length to be 4 meters more than twice the width. Write an equation for the perimeter in terms of the width w, then solve for w to find the garden's dimensions. Answer: w = 8 meters, length = 20 meters Solution: When solving geometry problems with unknown dimensions, we often use variables to represent the measurements. The perimeter of a rectangle is calculated using a specific formula.
   Full step-by-step solution

   When solving geometry problems with unknown dimensions, we often use variables to represent the measurements. The perimeter of a rectangle is calculated using a specific formula. If a relationship exists between length and width, we can express one in terms of the other. This allows us to create a single equation with one variable that we can solve using algebraic methods like combining like terms and isolating the variable.