Complex Expressions

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Answer Key & Explanations

Complex Expressions · Grade 9 · Worksheet 2

1. Mason is analyzing the expression 1500(0.92)^t that models the value of his car over time. In this expression, what does the number 0.92 represent? Answer: 0.92 represents the decay factor per time period, meaning the car retains 92% of its value each year (8% annual depreciation rate). Solution: The expression 1500(0.92)^t represents exponential decay, where 1500 is the initial value of the car. The base of the exponent, 0.92, is the decay factor.
   Full step-by-step solution

   Step 1: The expression 1500(0.92)^t represents exponential decay, where 1500 is the initial value of the car. Step 2: The base of the exponent, 0.92, is the decay factor. Step 3: A decay factor of 0.92 means the value decreases by 8% each time period (since 0.92 = 1 - 0.08). Step 4: Each year, the car's value multiplies by 0.92, which represents 92% of the previous year's value. Step 5: Therefore, 0.92 represents the multiplier applied to the car's value each year, corresponding to an 8% annual depreciation rate.

2. Emma is analyzing the trajectory of a rocket launch using quadratic functions. The rocket's height above ground is modeled by h(t) = -4.9t² + 98t + 20, where h is height in meters and t is time in seconds. At what time does the rocket reach its maximum height? Answer: 10 Solution: The rocket's height is modeled by h(t) = -4.9t² + 98t + 20 For a quadratic function in the form ax² + bx + c, the vertex (maximum point) occurs at t = -b/(2a) Identify a = -4.9 and b = 98 Calculate t = -98/(2 × -4.9) = -98/(-9.8) = 10 The rocket reaches its maximum height at t = 10 seconds The…
   Full step-by-step solution

   Step 1: The rocket's height is modeled by h(t) = -4.9t² + 98t + 20 Step 2: For a quadratic function in the form ax² + bx + c, the vertex (maximum point) occurs at t = -b/(2a) Step 3: Identify a = -4.9 and b = 98 Step 4: Calculate t = -98/(2 × -4.9) = -98/(-9.8) = 10 Step 5: The rocket reaches its maximum height at t = 10 seconds The answer is 10.

3. Liam is analyzing the trajectory of a basketball shot. The ball's height above the ground is modeled by the function h(t) = -16t² + 24t + 6, where h is height in feet and t is time in seconds. At what time does the basketball reach its maximum height? Answer: 0.75 Solution: We are given the height function: h(t) = -16t² + 24t + 6.
   Full step-by-step solution

   We are given the height function: h(t) = -16t² + 24t + 6. This is a quadratic function in the form h(t) = at² + bt + c, where: a = -16 b = 24 c = 6 Since a is negative, the parabola opens downward, so the vertex of the parabola gives the maximum height. The t-coordinate of the vertex for a quadratic function at² + bt + c is given by: t = -b / (2a) Step 1: Identify b and a. b = 24 a = -16 Step 2: Plug into the vertex formula. t = -24 / (2 * -16) Step 3: Simplify the denominator. 2 * -16 = -32 Step 4: Write the fraction. t = -24 / (-32) Step 5: Simplify the signs. Negative divided by negative is positive: t = 24 / 32 Step 6: Reduce the fraction. Divide numerator and denominator by 8: 24 ÷ 8 = 3 32 ÷ 8 = 4 So t = 3/4 Step 7: Convert to decimal. 3/4 = 0.75 Therefore, the basketball reaches its maximum height at t = 0.75 seconds. Answer: 0.75

4. Mere is analyzing the profit of a small business that sells handmade scarves. The daily profit P (in dollars) is modeled by the expression P(n) = 450n - (150n + 400), where n is the number of scarves sold per day. Interpret the meaning of the term 150n and the constant 400 in the context of the business. Answer: 150n represents the variable cost of materials and labor per scarf sold; 400 represents the fixed daily operating costs (such as rent and utilities) that do not depend on the number of scarves sold. Solution: The expression P(n) = 450n - (150n + 400) can be rewritten as P(n) = 450n - 150n - 400, where 450n is the total revenue from selling n scarves at $450 each.
   Full step-by-step solution

   Step 1: The expression P(n) = 450n - (150n + 400) can be rewritten as P(n) = 450n - 150n - 400, where 450n is the total revenue from selling n scarves at $450 each. Step 2: The term 150n inside the parentheses represents the total cost that varies with the number of scarves produced, such as materials (yarn, dye) and direct labor per scarf. Since it is multiplied by n, it depends on how many scarves are made and sold. Step 3: The constant 400 inside the parentheses is a fixed cost that does not depend on n. This could include daily expenses like rent for the workshop, electricity, or insurance—costs the business must pay regardless of production. Step 4: Therefore, 150n is the variable cost per day for n scarves, and 400 is the fixed daily cost. The answer is: 150n represents the variable cost of materials and labor per scarf sold; 400 represents the fixed daily operating costs.

5. Liam is designing a rectangular garden with a perimeter of 60 meters. He wants the length to be 6 meters more than twice the width. Write an equation in terms of width w that represents this situation, then determine the dimensions of the garden. Answer: width = 8 meters, length = 22 meters Solution: We have a rectangular garden. Perimeter = 60 meters. Let width = w meters.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We have a rectangular garden. Perimeter = 60 meters. Let width = w meters. Length is 6 meters more than twice the width. So length = 2w + 6. --- **Step 2: Write the perimeter equation** Perimeter of a rectangle = 2 × length + 2 × width. So: 2 × (length) + 2 × (width) = 60 Substitute length = 2w + 6: 2 × (2w + 6) + 2 × w = 60 --- **Step 3: Simplify and solve for w** First, distribute the 2 in 2(2w + 6): 4w + 12 + 2w = 60 Combine like terms: 6w + 12 = 60 Subtract 12 from both sides: 6w = 48 Divide both sides by 6: w = 8 --- **Step 4: Find the length** Length = 2w + 6 = 2 × 8 + 6 = 16 + 6 = 22 --- **Step 5: Check** Perimeter = 2 × length + 2 × width = 2 × 22 + 2 × 8 = 44 + 16 = 60 ✅ Length 22 is indeed 6 more than twice the width (2 × 8 = 16, 16 + 6 = 22). --- **Final answer:** Width = 8 meters, Length = 22 meters

6. Noah is analyzing the expression 2400(1.06)^t that models the growth of his investment. In this expression, what does the number 1.06 represent? Answer: 1.06 represents the growth factor per time period, meaning the investment multiplies by 1.06 each year (6% annual growth rate). Solution: The expression 2400(1.06)^t is in the form A = P(1+r)^t, where P is the initial amount, r is the growth rate per time period, and t is the number of time periods. Here, 2400 is the initial investment (P).
   Full step-by-step solution

   Step 1: The expression 2400(1.06)^t is in the form A = P(1+r)^t, where P is the initial amount, r is the growth rate per time period, and t is the number of time periods. Step 2: Here, 2400 is the initial investment (P). Step 3: The base of the exponent is 1.06. This is the growth factor. Step 4: A growth factor of 1.06 means the investment is multiplied by 1.06 each year. Step 5: Since 1.06 = 1 + 0.06, the growth rate r is 0.06, which is 6%. Step 6: Therefore, 1.06 represents the factor by which the investment grows each year, corresponding to a 6% annual increase. The answer is: 1.06 represents the growth factor per time period, meaning the investment multiplies by 1.06 each year (6% annual growth rate).

7. Olivia is analyzing the path of a ball thrown into the air. The height of the ball, in meters, after t seconds is given by the expression -5(t - 3)^2 + 45. In this expression, interpret the meaning of the number 45 and the number -5 in the context of the ball's motion. Answer: 45 is the maximum height in meters; -5 indicates the parabola opens downward and relates to the acceleration due to gravity. Solution: Recognize the expression is in vertex form: a(t - h)^2 + k, where a = -5, h = 3, k = 45. In this context, t is time in seconds, and the expression gives height in meters.
   Full step-by-step solution

   Step 1: Recognize the expression is in vertex form: a(t - h)^2 + k, where a = -5, h = 3, k = 45. Step 2: In this context, t is time in seconds, and the expression gives height in meters. Step 3: The vertex (h, k) = (3, 45) represents the maximum point of the parabola because a is negative (opens downward). Step 4: Thus, k = 45 is the maximum height the ball reaches, which occurs at t = 3 seconds. Step 5: The coefficient a = -5 determines the shape: negative means the parabola opens downward (ball goes up then down), and the magnitude 5 relates to the effect of gravity (in simplified form, half of g ≈ 9.8 m/s^2, rounded). Final interpretation: 45 represents the maximum height of 45 meters; -5 indicates the ball's path is a downward-opening parabola due to gravity.

8. Olivia is analyzing the expression 2400(0.93)^t that models the value of her car over t years. In this expression, what does the number 0.93 represent? Answer: 0.93 represents the decay factor per year, meaning the car retains 93% of its value each year (a 7% annual depreciation rate). Solution: The expression 2400(0.93)^t represents exponential decay, where 2400 is the initial value of the car. Step 2: The base of the exponent, 0.93, is the decay factor.
   Full step-by-step solution

   Step 1: The expression 2400(0.93)^t represents exponential decay, where 2400 is the initial value of the car. Step 2: The base of the exponent, 0.93, is the decay factor. Step 3: A decay factor of 0.93 means the car's value is multiplied by 0.93 each year. Step 4: Since 0.93 = 1 - 0.07, this corresponds to a 7% decrease in value each year. Step 5: Therefore, 0.93 represents the fraction of the car's value that remains after one year, which is 93% of the previous year's value.