Rearrange Formulas

lessonbunny.com

Answer Key & Explanations

Rearrange Formulas · Grade 9 · Worksheet 1

1. A right circular cone has a height of 18 cm and a base radius of 8 cm. A horizontal cross-section is taken at a height of 12 cm from the vertex. What is the radius of this circular cross-section? Answer: 5.33 Solution: Identify the similar triangles formed by the cone's side profile. The full cone has height 18 cm and base radius 8 cm. The smaller cone from the vertex to the cross-section has height 12 cm.
   Full step-by-step solution

   Step 1: Identify the similar triangles formed by the cone's side profile. The full cone has height 18 cm and base radius 8 cm. The smaller cone from the vertex to the cross-section has height 12 cm. Step 2: Set up the proportion using similar triangles: (radius of cross-section) / (base radius) = (height of smaller cone) / (total height) Step 3: Substitute the known values: r / 8 = 12 / 18 Step 4: Simplify the right side: 12/18 = 2/3 Step 5: Solve for r: r = 8 × (2/3) = 16/3 = 5.333... Step 6: Round to two decimal places: r = 5.33 cm The answer is 5.33.

2. Rearrange V = (1/3)πr²h to solve for r Answer: r = √(3V/(πh)) 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 πh to isolate r²: r² = 3V/(πh) Take the square root of both sides: r = √(3V/(πh)) The rearranged formula is r = √(3V/(πh))
   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 πh to isolate r²: r² = 3V/(πh) Step 4: Take the square root of both sides: r = √(3V/(πh)) The rearranged formula is r = √(3V/(πh))

3. Rearrange A = 2πr² + 7πrh to solve for h. Answer: h = (A - 2πr²) / (7πr) Solution: Start with A = 2πr² + 7πrh. Subtract 2πr² from both sides: A - 2πr² = 7πrh. Divide both sides by 7πr: (A - 2πr²) / (7πr) = h.
   Full step-by-step solution

   Step 1: Start with A = 2πr² + 7πrh. Step 2: Subtract 2πr² from both sides: A - 2πr² = 7πrh. Step 3: Divide both sides by 7πr: (A - 2πr²) / (7πr) = h. Step 4: Write the final answer: h = (A - 2πr²) / (7πr).

4. 2x² + 5x - 3 = 0 Answer: x = 0.5, x = -3 Solution: We are solving the quadratic equation: 2x² + 5x - 3 = 0. The equation is in the form ax² + bx + c = 0. Here: a = 2, b = 5, c = -3.
   Full step-by-step solution

   We are solving the quadratic equation: 2x² + 5x - 3 = 0. Step 1: Identify coefficients The equation is in the form ax² + bx + c = 0. Here: a = 2, b = 5, c = -3. Step 2: Check if factoring is possible We look for two numbers that multiply to a * c = 2 * (-3) = -6, and add to b = 5. Possible pairs for -6: (1, -6) sum = -5, (-1, 6) sum = 5. Yes, -1 and 6 work. Step 3: Rewrite the middle term using these numbers 2x² + 6x - 1x - 3 = 0. Step 4: Factor by grouping Group terms: (2x² + 6x) + (-1x - 3) = 0. Factor each group: 2x(x + 3) - 1(x + 3) = 0. Step 5: Factor out the common factor (x + 3)(2x - 1) = 0. Step 6: Apply the zero product property If (x + 3)(2x - 1) = 0, then: x + 3 = 0 or 2x - 1 = 0. Step 7: Solve each equation From x + 3 = 0: x = -3. From 2x - 1 = 0: 2x = 1, x = 1/2 = 0.5. Final answer: x = 0.5, x = -3.

5. Rearrange S = 12πr² + 9πrh to solve for h. Answer: h = (S - 12πr²) / (9πr) Solution: Start with S = 12πr² + 9πrh. Subtract 12πr² from both sides: S - 12πr² = 9πrh. Divide both sides by 9πr: (S - 12πr²) / (9πr) = h.
   Full step-by-step solution

   Step 1: Start with S = 12πr² + 9πrh. Step 2: Subtract 12πr² from both sides: S - 12πr² = 9πrh. Step 3: Divide both sides by 9πr: (S - 12πr²) / (9πr) = h. Step 4: Write the final answer: h = (S - 12πr²) / (9πr).

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. 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 of the garden is 20 - w meters, write a formula for the area A as a function of the width w. Then, determine the maximum possible area of the garden. Answer: 100 Solution: - Perimeter = 40 m - Width = w - Length = 20 - w - Need area A as a function of w, then find maximum area.
   Full step-by-step solution

   Step 1: Understand the problem We are told: - Perimeter = 40 m - Width = w - Length = 20 - w - Need area A as a function of w, then find maximum area. --- Step 2: Write the area formula Area of a rectangle = length × width So: A(w) = (20 - w) × w A(w) = 20w - w^2 --- Step 3: Recognize the type of function A(w) = -w^2 + 20w is a quadratic function (parabola) opening downwards (since coefficient of w^2 is negative). Maximum occurs at the vertex. --- Step 4: Find the vertex For a quadratic in form f(x) = ax^2 + bx + c, vertex at x = -b/(2a). Here: a = -1, b = 20 w = -20 / (2 × -1) = -20 / -2 = 10 So width for maximum area is w = 10 m. --- Step 5: Find length when w = 10 Length = 20 - w = 20 - 10 = 10 m. --- Step 6: Compute maximum area Area = length × width = 10 × 10 = 100 m^2. --- Step 7: Conclusion The maximum possible area is 100 square meters.