Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 3

1. A technology company is modeling the growth of its user base. The number of users U(t) after t months is given by the exponential function U(t) = 1200 × 2^(t/6). How many months will it take for the user base to reach 19,200 users? Answer: 24 Solution: Set up the equation: 1200 × 2^(t/6) = 19200 Divide both sides by 1200: 2^(t/6) = 16 Recognize that 16 = 2^4, so 2^(t/6) = 2^4 Set the exponents equal: t/6 = 4 Multiply both sides by 6: t = 24 It will take 24 months for the user base to reach 19,200 users.
   Full step-by-step solution

   Step 1: Set up the equation: 1200 × 2^(t/6) = 19200 Step 2: Divide both sides by 1200: 2^(t/6) = 16 Step 3: Recognize that 16 = 2^4, so 2^(t/6) = 2^4 Step 4: Set the exponents equal: t/6 = 4 Step 5: Multiply both sides by 6: t = 24 Step 6: It will take 24 months for the user base to reach 19,200 users.

2. Noah is saving money to buy a new laptop that costs $1,216. He currently has $256 saved and plans to save $48 each week from his part-time job. Write an equation to represent the number of weeks (w) Noah needs to save before he can afford the laptop, then determine how many weeks it will take. Answer: 20 Solution: Let w represent the number of weeks Noah saves. Noah saves $48 each week, so after w weeks he saves 48w dollars. He starts with $256, so total savings = 256 + 48w.
   Full step-by-step solution

   Step 1: Let w represent the number of weeks Noah saves. Step 2: Noah saves $48 each week, so after w weeks he saves 48w dollars. Step 3: He starts with $256, so total savings = 256 + 48w. Step 4: Set this equal to the laptop cost: 256 + 48w = 1216. Step 5: Subtract 256 from both sides: 48w = 1216 - 256 = 960. Step 6: Divide both sides by 48: w = 960 / 48 = 20. Step 7: Verify: 256 + 48(20) = 256 + 960 = 1216. Correct. The answer is 20 weeks.

3. A company's profit in thousands of dollars is modeled by the function P(t) = 2t^2 - 12t + 16, where t is the number of years after 2020. During what year will the company's profit reach its minimum value? Answer: 2023 Solution: The profit function is P(t) = 2t^2 - 12t + 16, which is a quadratic function opening upward (since the coefficient of t^2 is positive). The minimum profit occurs at the vertex of the parabola.
   Full step-by-step solution

   Step 1: The profit function is P(t) = 2t^2 - 12t + 16, which is a quadratic function opening upward (since the coefficient of t^2 is positive). Step 2: The minimum profit occurs at the vertex of the parabola. For a quadratic in the form ax^2 + bx + c, the t-coordinate of the vertex is given by t = -b/(2a). Step 3: Here, a = 2 and b = -12, so t = -(-12)/(2*2) = 12/4 = 3. Step 4: Since t represents years after 2020, the minimum profit occurs 3 years after 2020. Step 5: The year is 2020 + 3 = 2023. The answer is 2023.

4. A drone is flying along a parabolic path modeled by the equation h(t) = -2t² + 16t + 20, where h represents the height in meters and t represents time in seconds. Determine the maximum height the drone reaches and at what time this occurs. Answer: 52 Solution: The quadratic function is h(t) = -2t² + 16t + 20 Since the coefficient of t² is negative (-2), the parabola opens downward, so the vertex represents the maximum height The t-coordinate of the vertex is found using t = -b/(2a) t = -16/(2×-2) = -16/-4 = 4 seconds Substitute t = 4 into the original…
   Full step-by-step solution

   Step 1: The quadratic function is h(t) = -2t² + 16t + 20 Step 2: Since the coefficient of t² is negative (-2), the parabola opens downward, so the vertex represents the maximum height Step 3: The t-coordinate of the vertex is found using t = -b/(2a) Step 4: t = -16/(2×-2) = -16/-4 = 4 seconds Step 5: Substitute t = 4 into the original equation to find the maximum height Step 6: h(4) = -2(4)² + 16(4) + 20 = -2(16) + 64 + 20 = -32 + 64 + 20 = 52 meters Step 7: The drone reaches its maximum height of 52 meters at 4 seconds Step 8: The question asks for the maximum height, which is 52 meters

5. Charlotte is designing a rectangular window. The length of the window is 7 feet more than twice its width. The diagonal of the window is 22 feet. Write an equation in one variable that can be used to find the width of the window. Answer: x^2 + (2x + 7)^2 = 22^2 Solution: Let x represent the width of the window in feet. Step 2: The length is 7 feet more than twice the width, so length = 2x + 7. Step 3: The diagonal, length, and width form a right triangle.
   Full step-by-step solution

   Step 1: Let x represent the width of the window in feet. Step 2: The length is 7 feet more than twice the width, so length = 2x + 7. Step 3: The diagonal, length, and width form a right triangle. By the Pythagorean theorem, (width)^2 + (length)^2 = (diagonal)^2. Step 4: Substitute the expressions: x^2 + (2x + 7)^2 = 22^2. This is the required equation in one variable.

6. Emma is designing a rectangular garden with an area of 135 square meters. The length is 5 meters more than twice the width. Write an equation to find the width. Answer: w(2w + 5) = 135 Solution: The length is 5 more than twice the width, so length = 2w + 5 Area = length × width = (2w + 5) × w The area is given as 135 square meters, so w(2w + 5) = 135 The equation is w(2w + 5) = 135
   Full step-by-step solution

   Step 1: Let w represent the width of the garden Step 2: The length is 5 more than twice the width, so length = 2w + 5 Step 3: Area = length × width = (2w + 5) × w Step 4: The area is given as 135 square meters, so w(2w + 5) = 135 The equation is w(2w + 5) = 135

7. A scientist is modeling the population growth of a bacterial colony using the function P(t) = 500 * 2^(t/3), where t is time in hours. After how many hours will the population reach 16,000 bacteria? Answer: 15 Solution: Set up the equation: 500 * 2^(t/3) = 16,000 Divide both sides by 500: 2^(t/3) = 32 Recognize that 32 = 2^5, so 2^(t/3) = 2^5 Set the exponents equal: t/3 = 5 Multiply both sides by 3: t = 15 The population will reach 16,000 bacteria after 15 hours.
   Full step-by-step solution

   Step 1: Set up the equation: 500 * 2^(t/3) = 16,000 Step 2: Divide both sides by 500: 2^(t/3) = 32 Step 3: Recognize that 32 = 2^5, so 2^(t/3) = 2^5 Step 4: Set the exponents equal: t/3 = 5 Step 5: Multiply both sides by 3: t = 15 Step 6: The population will reach 16,000 bacteria after 15 hours.

8. Matiu is building a rectangular skateboard ramp. The length of the ramp's base is 4 meters longer than twice its width. The area of the rectangular base is 70 square meters. Write an equation in one variable to represent this situation and find the width of the base. Answer: 5 Solution: Let w represent the width of the base in meters. The length is 4 meters longer than twice the width, so length = 2w + 4. Area of a rectangle = length × width = 70.
   Full step-by-step solution

   Step 1: Let w represent the width of the base in meters. Step 2: The length is 4 meters longer than twice the width, so length = 2w + 4. Step 3: Area of a rectangle = length × width = 70. Step 4: Write the equation: w(2w + 4) = 70. Step 5: Expand: 2w^2 + 4w = 70. Step 6: Subtract 70 from both sides: 2w^2 + 4w - 70 = 0. Step 7: Divide the entire equation by 2: w^2 + 2w - 35 = 0. Step 8: Factor the quadratic: (w + 7)(w - 5) = 0. Step 9: Set each factor equal to zero: w + 7 = 0 or w - 5 = 0, giving w = -7 or w = 5. Step 10: Width cannot be negative, so w = 5. The width of the base is 5 meters.