Create Equations

lessonbunny.com

Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 2

1. Aroha is designing a rectangular banner with perimeter 120 cm. The length is 4x + 6 cm and the width is 2x - 4 cm. Write an equation for x. Answer: 2(4x + 6) + 2(2x - 4) = 120 Solution: Step 1: Write the perimeter formula for a rectangle: P = 2L + 2W Step 2: Substitute the given values: P = 120, L = 4x + 6, W = 2x - 4 Step 3: Plug into the formula: 2(4x + 6) + 2(2x - 4) = 120 Step 4: This is the equation that can be solved for x Step 5: To verify, we can simplify: 8x + 12 + 4x…
   Full step-by-step solution

   Step 1: Write the perimeter formula for a rectangle: P = 2L + 2W Step 2: Substitute the given values: P = 120, L = 4x + 6, W = 2x - 4 Step 3: Plug into the formula: 2(4x + 6) + 2(2x - 4) = 120 Step 4: This is the equation that can be solved for x Step 5: To verify, we can simplify: 8x + 12 + 4x - 8 = 120 → 12x + 4 = 120 → 12x = 116 → x = 116/12 = 29/3 The equation is 2(4x + 6) + 2(2x - 4) = 120

2. Mason is designing a rectangular garden. The length of the garden is 7 meters more than the width. The area of the garden is 294 square meters. Write an equation to find the width of the garden. Answer: w(w + 7) = 294 Solution: Let w represent the width of the garden in meters. The length is 7 meters more than the width, so length = w + 7. The area of a rectangle is given by A = length × width.
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 7 meters more than the width, so length = w + 7. Step 3: The area of a rectangle is given by A = length × width. Step 4: Substitute the expressions: (w + 7) × w = 294. Step 5: The equation is w(w + 7) = 294.

3. A rectangular prism has a length that is 9 cm longer than its width, and a height that is half its width. The volume of the prism is 972 cubic centimeters. Write an equation in one variable to represent this situation, using x for the width of the prism. Answer: x(x+9)(x/2) = 972 Solution: Define the variable. Let x be the width of the rectangular prism in centimeters. Length = x + 9 (since length is 9 cm longer than width) Height = x / 2 (since height is half the width) Recall the formula for volume of a rectangular prism: Volume = length * width * height.
   Full step-by-step solution

   Step 1: Define the variable. Let x be the width of the rectangular prism in centimeters. Step 2: Express the other dimensions in terms of x. Length = x + 9 (since length is 9 cm longer than width) Height = x / 2 (since height is half the width) Step 3: Recall the formula for volume of a rectangular prism: Volume = length * width * height. Step 4: Substitute the expressions: Volume = (x + 9) * x * (x / 2). Step 5: Set this equal to the given volume of 972 cubic centimeters. The equation is: x * (x + 9) * (x / 2) = 972. This can also be written as: x(x+9)(x/2) = 972.

4. Olivia is designing a rectangular banner with an area of 315 square inches. The length is 7 inches more than the width. Write an equation to find the width. Answer: w(w+7)=315 Solution: The length is 7 inches more than the width, so length = w + 7 Area of rectangle = length × width Substitute the expressions: (w + 7) × w = 315 Write the equation: w(w + 7) = 315 This is the equation that can be used to find the width.
   Full step-by-step solution

   Step 1: Let w represent the width of the banner Step 2: The length is 7 inches more than the width, so length = w + 7 Step 3: Area of rectangle = length × width Step 4: Substitute the expressions: (w + 7) × w = 315 Step 5: Write the equation: w(w + 7) = 315 This is the equation that can be used to find the width.

5. A right circular cone is inscribed in a sphere such that the vertex and all points on the circular base lie on the sphere's surface. The sphere has a radius of 10 cm, and the height of the cone is 16 cm. What is the volume of the cone? Answer: 256π Solution: When a solid is inscribed in a sphere, key points of the solid touch the sphere's surface.
   Full step-by-step solution

   When a solid is inscribed in a sphere, key points of the solid touch the sphere's surface. For a right circular cone in a sphere, understanding the spatial relationship between the cone's dimensions and the sphere's radius is crucial. The distance from the sphere's center to the cone's vertex and the radius of the cone's base can be determined using geometric principles involving right triangles formed by these elements.

6. A company's profit P(x) from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. What is the maximum number of units the company can sell before they start losing money? Answer: 50 Solution: The company starts losing money when profit becomes zero, so set P(x) = 0 -2x² + 120x - 1000 = 0 Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Solve the quadratic equation using factoring: (x - 10)(x - 50) = 0 The solutions are x = 10 and x = 50 Since this is a…
   Full step-by-step solution

   Step 1: The company starts losing money when profit becomes zero, so set P(x) = 0 Step 2: -2x² + 120x - 1000 = 0 Step 3: Divide the entire equation by -2 to simplify: x² - 60x + 500 = 0 Step 4: Solve the quadratic equation using factoring: (x - 10)(x - 50) = 0 Step 5: The solutions are x = 10 and x = 50 Step 6: Since this is a downward-opening parabola (coefficient of x² is negative), the profit is positive between the roots Step 7: The company makes profit when 10 < x < 50, and loses money when x > 50 Step 8: Therefore, the maximum number of units before losing money is 50 The answer is 50.

7. Kaia is saving money to buy a new laptop that costs $1,200. She already has $240 saved and decides to save a fixed amount each week from her part-time job. After 12 weeks of saving, she still needs $360 to reach her goal. Write an equation in one variable to represent this situation, where x represents the amount Kaia saves each week. Do not solve the equation. Answer: 240 + 12x + 360 = 1200 or 240 + 12x = 840 or equivalent Solution: Identify the known and unknown values. Kaia's goal is $1200. She already has $240.
   Full step-by-step solution

   Step 1: Identify the known and unknown values. Kaia's goal is $1200. She already has $240. After saving for 12 weeks, she still needs $360. Let x = amount saved each week. Step 2: Express the total amount saved after 12 weeks. Amount saved after 12 weeks = 240 + 12x Step 3: Write an equation showing that the amount saved plus the amount still needed equals the goal. (240 + 12x) + 360 = 1200 Step 4: Simplify if desired. 240 + 12x + 360 = 1200 600 + 12x = 1200 or equivalently, 240 + 12x = 840 The required equation is 240 + 12x + 360 = 1200 (or any equivalent form).