Complex Expressions

lessonbunny.com

Answer Key & Explanations

Complex Expressions · Grade 9 · Worksheet 1

1. Tane is analyzing the profit from a school fundraising car wash. The profit P(x) in dollars for washing x cars is given by the expression P(x) = 12x - 75. In this expression, what does the number 12 represent and what does the number -75 represent in the context of the car wash? Answer: 12 represents the amount of money earned per car washed ($12 per car), and -75 represents the fixed costs or initial expenses ($75) that must be paid before any profit is made. Solution: The expression P(x) = 12x - 75 is in the form of a linear function where x is the number of cars washed. The coefficient 12 is multiplied by x.
   Full step-by-step solution

   Step 1: The expression P(x) = 12x - 75 is in the form of a linear function where x is the number of cars washed. Step 2: The coefficient 12 is multiplied by x. For each additional car washed (x increases by 1), the profit increases by 12 dollars. This means the car wash earns $12 per car washed. Step 3: The constant term -75 is the value of P(0), since P(0) = 12(0) - 75 = -75. This represents the profit when zero cars are washed, which is a loss of $75. This $75 is the fixed cost (such as soap, sponges, and advertising) that must be paid regardless of how many cars are washed. Step 4: Therefore, in this context, 12 represents the revenue earned per car washed ($12 per car), and -75 represents the fixed costs of $75 that must be covered before making a profit. The answer: 12 represents the amount of money earned per car washed ($12 per car), and -75 represents the fixed costs or initial expenses ($75).

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 and solve a system of equations to determine the dimensions of Liam's garden. Answer: width = 6 meters, length = 14 meters Solution: The perimeter formula for rectangles creates one equation, while verbal descriptions of how dimensions relate create additional equations.
   Full step-by-step solution

   Many real-world geometry problems involve setting up equations based on given relationships between dimensions. The perimeter formula for rectangles creates one equation, while verbal descriptions of how dimensions relate create additional equations. Solving such systems helps find unknown measurements in design and construction scenarios.

3. Noah is analyzing the expression 2500(0.91)^t that models the value of a car after t years. In this expression, what does the number 0.91 represent? Answer: 0.91 represents the decay factor per year, meaning the car retains 91% of its value each year (a 9% annual depreciation rate). Solution: The expression 2500(0.91)^t represents exponential decay, where 2500 is the initial value of the car. The base of the exponent, 0.91, is the decay factor.
   Full step-by-step solution

   Step 1: The expression 2500(0.91)^t represents exponential decay, where 2500 is the initial value of the car. Step 2: The base of the exponent, 0.91, is the decay factor. Step 3: A decay factor of 0.91 means the car retains 91% of its value each year (since 0.91 = 91/100). Step 4: The annual depreciation rate is 100% - 91% = 9%. Step 5: Therefore, 0.91 represents the multiplier applied to the car's value each year, corresponding to a 9% annual decrease in value.

4. Charlotte is analyzing the profit of a small business that sells handmade candles. The monthly profit P (in dollars) is modeled by the expression 12n(50 - n) - 200, where n is the number of candle designs offered. Interpret the meaning of the factor (50 - n) in the expression, considering the context of the business. Answer: The factor (50 - n) represents the number of candle designs that are not being offered, or equivalently, the remaining capacity for new designs up to a maximum of 50 designs. In context, it indicates that as n increases, the term (50 - n) decreases, showing a diminishing return on offering more designs due to market saturation or resource limits. Solution: Identify the expression and its parts. The profit expression is 12n(50 - n) - 200.
   Full step-by-step solution

   Step 1: Identify the expression and its parts. The profit expression is 12n(50 - n) - 200. The variables and constants are: 12 (profit per design in dollars), n (number of designs offered), 50 (maximum possible designs), (50 - n) (difference between maximum and actual designs), 200 (fixed monthly costs in dollars). Step 2: Interpret the factor (50 - n). The number 50 represents the maximum number of candle designs the business can feasibly offer, perhaps due to market demand or production capacity. The term (50 - n) calculates how many designs are not being offered, i.e., the gap between the maximum and the current number. Step 3: Understand the effect on profit. When n is small, (50 - n) is large, contributing to higher profit. As n approaches 50, (50 - n) becomes small, reducing the product 12n(50 - n) and thus profit. This models diminishing returns: offering too many designs may oversaturate the market or strain resources, decreasing overall profit. Step 4: Conclude the interpretation. The factor (50 - n) represents the number of designs not offered, acting as a constraint that reduces profit as n increases toward the maximum of 50. The answer is: The factor (50 - n) represents the number of candle designs not being offered, up to a maximum of 50 designs.

5. 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 an equation in terms of the width w that represents this situation, then determine the dimensions of the garden. Answer: w = 6, length = 16 Solution: The key is translating the word description into mathematical expressions and then using the known total to solve for the unknown values.
   Full step-by-step solution

   Many real-world geometry problems involve setting up equations where one quantity depends on another. For example, if you know the total distance around a shape and how two dimensions relate to each other, you can use algebraic expressions to represent this relationship. The key is translating the word description into mathematical expressions and then using the known total to solve for the unknown values.

6. Mason is analyzing the expression 3200(0.97)^t that models the value of his car after t years. In this expression, what does the number 0.97 represent? Answer: 0.97 represents the decay factor per year, meaning the car retains 97% of its value each year (a 3% annual depreciation rate). Solution: The expression 3200(0.97)^t represents exponential decay, where 3200 is the initial value of the car. Step 2: The base of the exponent, 0.97, is the decay factor.
   Full step-by-step solution

   Step 1: The expression 3200(0.97)^t represents exponential decay, where 3200 is the initial value of the car. Step 2: The base of the exponent, 0.97, is the decay factor. Step 3: A decay factor of 0.97 means the car's value is multiplied by 0.97 each year. Step 4: Since 0.97 = 1 - 0.03, this corresponds to a 3% decrease in value per year. Step 5: Therefore, 0.97 represents the factor by which the car's value changes each year, indicating it retains 97% of its previous year's value (a 3% annual depreciation rate).

7. √(16) + 3² - 2³ = ? Answer: 5 Solution: √(16) + 3² - 2³ √(16) means the principal (positive) square root of 16.
   Full step-by-step solution

   Let's solve step by step. We have: √(16) + 3² - 2³ **Step 1: Evaluate the square root** √(16) means the principal (positive) square root of 16. √(16) = 4 **Step 2: Evaluate the exponent 3²** 3² = 3 × 3 = 9 **Step 3: Evaluate the exponent 2³** 2³ = 2 × 2 × 2 = 8 **Step 4: Substitute back into the expression** 4 + 9 - 8 **Step 5: Perform addition and subtraction from left to right** 4 + 9 = 13 13 - 8 = 5 **Final Answer:** 5

8. Given the function f(x) = 2x² - 5x + 3 and g(x) = x - 1, find the value of f(g(4)). Answer: 6 Solution: f(x) = 2x² - 5x + 3 g(x) = x - 1 We need to find f(g(4)). Find g(4) g(x) = x - 1 So g(4) = 4 - 1 = 3 Substitute g(4) into f(x) We now need f(g(4)) = f(3).
   Full step-by-step solution

   We are given: f(x) = 2x² - 5x + 3 g(x) = x - 1 We need to find f(g(4)). --- **Step 1: Find g(4)** g(x) = x - 1 So g(4) = 4 - 1 = 3 --- **Step 2: Substitute g(4) into f(x)** We now need f(g(4)) = f(3). f(x) = 2x² - 5x + 3 So f(3) = 2*(3)² - 5*(3) + 3 --- **Step 3: Calculate f(3)** First, 3² = 9 Then 2*(9) = 18 Next, -5*(3) = -15 So f(3) = 18 - 15 + 3 --- **Step 4: Simplify** 18 - 15 = 3 3 + 3 = 6 --- **Final Answer:** 6

9. Mason is analyzing the expression 2500(0.92)^t that models the depreciation of his car's value. 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 2500(0.92)^t represents exponential decay, where 2500 is the initial value of the car. Step 2: The base of the exponent, 0.92, is the decay factor.
   Full step-by-step solution

   Step 1: The expression 2500(0.92)^t represents exponential decay, where 2500 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 car retains 92% of its value each time period (since 0.92 = 1 - 0.08). Step 4: Each year, the car's value multiplies by 0.92, which represents 100% of the previous value minus an 8% loss. Step 5: Therefore, 0.92 represents the multiplier applied to the car's value each year, corresponding to an 8% annual depreciation rate.