Create Equations

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Answer Key & Explanations

Create Equations · Grade 9 · Worksheet 3

1. A scientist is modeling bacterial growth in a lab culture. The initial population is 500 bacteria, and it doubles every 3 hours. Write an exponential function P(t) = a * b^t that models the population after t hours, and determine how many bacteria will be present after 9 hours. Answer: 4000 Solution: Identify the initial population: a = 500 Determine the growth factor: Since the population doubles every 3 hours, the base b = 2 Account for the doubling period: The exponent needs to reflect that doubling happens every 3 hours, so we use t/3 in the exponent Write the function: P(t) = 500 *…
   Full step-by-step solution

   Step 1: Identify the initial population: a = 500 Step 2: Determine the growth factor: Since the population doubles every 3 hours, the base b = 2 Step 3: Account for the doubling period: The exponent needs to reflect that doubling happens every 3 hours, so we use t/3 in the exponent Step 4: Write the function: P(t) = 500 * 2^(t/3) Step 5: Calculate population after 9 hours: P(9) = 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The answer is 4000 bacteria.

2. 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 a system of equations to model this situation and solve for the dimensions of the garden. Answer: width = 6 meters, length = 14 meters Solution: This type of problem involves creating equations from geometric relationships. The perimeter formula for rectangles connects the total distance around with both dimensions.
   Full step-by-step solution

   This type of problem involves creating equations from geometric relationships. The perimeter formula for rectangles connects the total distance around with both dimensions. When one dimension depends algebraically on the other, substitution allows you to solve for both variables systematically.

3. Isabella is designing a rectangular garden with a perimeter of 42 meters. The length is 7 meters more than the width. Write an equation that relates the length (l) and width (w) of the garden. Answer: 2l + 2w = 42, l = w + 7 Solution: The perimeter of a rectangle is given by P = 2l + 2w, where l is length and w is width. We know the perimeter is 42 meters, so 2l + 2w = 42. The problem states the length is 7 meters more than the width, so l = w + 7.
   Full step-by-step solution

   Step 1: The perimeter of a rectangle is given by P = 2l + 2w, where l is length and w is width. Step 2: We know the perimeter is 42 meters, so 2l + 2w = 42. Step 3: The problem states the length is 7 meters more than the width, so l = w + 7. Step 4: Therefore, the two equations are: 2l + 2w = 42 and l = w + 7.

4. Noah is drawing a rectangular prism on a coordinate grid. The base of the prism is a rectangle with vertices at (1, 6), (6, 6), (6, 1), and (1, 1). The height of the prism is represented by the variable h. Write an equation in two variables for the total surface area, S, of the rectangular prism. Answer: S = 2lw + 2lh + 2wh or S = 2(5)(5) + 2(5)h + 2(5)h = 50 + 20h Solution: Find the dimensions of the base rectangle. The vertices are (1,6), (6,6), (6,1), and (1,1). The length (horizontal distance) is from x=1 to x=6, which is 5 units.
   Full step-by-step solution

   Step 1: Find the dimensions of the base rectangle. The vertices are (1,6), (6,6), (6,1), and (1,1). The length (horizontal distance) is from x=1 to x=6, which is 5 units. The width (vertical distance) is from y=1 to y=6, which is 5 units. So, the base is a 5 by 5 square. Let l = 5 and w = 5. Step 2: The height of the prism is h (the variable). Step 3: The surface area S of a rectangular prism is given by the formula: S = 2lw + 2lh + 2wh. Step 4: Substitute l = 5 and w = 5 into the formula: S = 2(5)(5) + 2(5)h + 2(5)h. Step 5: Simplify the equation: S = 2(25) + 10h + 10h = 50 + 20h. The final equation is S = 50 + 20h.

5. Aisha is analyzing the growth of bacteria in a lab experiment. The bacteria population doubles every 3 hours. If she starts with 500 bacteria, which function models the population P after t hours? Answer: A. P(t) = 500 * 2^(t/3) Solution: Step 1: Identify the initial population: 500 bacteria Step 2: The population doubles every 3 hours, so the growth factor is 2 Step 3: The number of doubling periods in t hours is t/3 Step 4: The exponential growth model is: initial amount × (growth factor)^(number of periods) Step 5: Substitute…
   Full step-by-step solution

   Step 1: Identify the initial population: 500 bacteria Step 2: The population doubles every 3 hours, so the growth factor is 2 Step 3: The number of doubling periods in t hours is t/3 Step 4: The exponential growth model is: initial amount × (growth factor)^(number of periods) Step 5: Substitute the values: P(t) = 500 × 2^(t/3) Step 6: Verify: After 3 hours (t=3), P(3) = 500 × 2^(3/3) = 500 × 2^1 = 1000 (doubled) After 6 hours (t=6), P(6) = 500 × 2^(6/3) = 500 × 2^2 = 2000 (doubled again) The correct function is P(t) = 500 × 2^(t/3)

6. Olivia is designing a rectangular garden with a perimeter of 50 meters. If the length is 5 meters more than twice the width, write an equation in two variables that represents this situation. Answer: P = 2l + 2w, l = 2w + 5 Solution: When creating equations from geometric contexts, identify the standard formulas involved and then incorporate the given relationships between variables.
   Full step-by-step solution

   When creating equations from geometric contexts, identify the standard formulas involved and then incorporate the given relationships between variables. For example, if a rectangle's area is fixed and one dimension is a multiple of the other, you would use the area formula and substitute the relationship.

7. Sophia is designing a rectangular garden with a perimeter of 36 meters. If the length is 6 meters more than the width, write an equation in two variables representing this situation. Answer: 2l + 2w = 36, l = w + 6 Solution: Let l represent the length and w represent the width. The perimeter formula for a rectangle is P = 2l + 2w. Given the perimeter is 36 meters, we write: 2l + 2w = 36.
   Full step-by-step solution

   Step 1: Let l represent the length and w represent the width. Step 2: The perimeter formula for a rectangle is P = 2l + 2w. Step 3: Given the perimeter is 36 meters, we write: 2l + 2w = 36. Step 4: The length is 6 meters more than the width, so: l = w + 6. Step 5: The system of equations is: 2l + 2w = 36 and l = w + 6.

8. Isabella is designing a rectangular garden with a perimeter of 72 meters. The length is 7 meters more than twice the width. Write an equation in terms of length (L) and width (W) that represents the perimeter. Answer: 2L + 2W = 72 Solution: The perimeter of a rectangle is calculated by adding twice the length and twice the width.
   Full step-by-step solution

   The perimeter of a rectangle is calculated by adding twice the length and twice the width. When given a relationship between dimensions, you can create equations that connect these variables to represent the total perimeter.