Interpret Expressions

lessonbunny.com

Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 1

1. Emma's business profit is modeled by P(x) = 7x² - 13x + 9 where x represents hundreds of units sold. What does the constant term 9 represent in this context? Answer: 9 Solution: In the profit function P(x) = 7x² - 13x + 9, the variable x represents hundreds of units sold. The constant term 9 is independent of x, meaning it doesn't change based on units sold.
   Full step-by-step solution

   Step 1: In the profit function P(x) = 7x² - 13x + 9, the variable x represents hundreds of units sold. Step 2: The constant term 9 is independent of x, meaning it doesn't change based on units sold. Step 3: In business profit functions, constant terms typically represent fixed costs or initial investments that must be paid regardless of sales. Step 4: Since this is a profit function and the constant is positive, it likely represents initial capital or fixed revenue sources. Step 5: The constant term 9 represents the profit (in appropriate monetary units) when zero units are sold. The answer is 9.

2. Isabella's company profit is modeled by P(x) = 12x² - 25x + 18, where x represents hundreds of units sold. What does the constant term 18 represent in this context? Answer: 18 Solution: In the profit function P(x) = 12x² - 25x + 18, the variable x represents hundreds of units sold. The constant term 18 is independent of x, meaning it does not change based on the number of units sold.
   Full step-by-step solution

   Step 1: In the profit function P(x) = 12x² - 25x + 18, the variable x represents hundreds of units sold. Step 2: The constant term 18 is independent of x, meaning it does not change based on the number of units sold. Step 3: In a profit function, constant terms typically represent fixed costs or initial values that apply regardless of sales. Step 4: To find what it represents, evaluate P(0): P(0) = 12(0)² - 25(0) + 18 = 18. Step 5: This means when zero units are sold, the profit is 18. Since no sales generate no revenue, this 18 must represent the initial profit or a fixed revenue source (such as a subsidy or investment) that exists even with no sales. The answer is 18.

3. Noah's company profit is modeled by P(x) = 9x² - 17x + 14 where x represents hundreds of units sold. What does the constant term 14 represent in this context? Answer: 14 Solution: The profit function is P(x) = 9x² - 17x + 14, where x is the number of hundreds of units sold. The constant term is 14, which does not change when x changes. When x = 0 (no units sold), P(0) = 9(0)² - 17(0) + 14 = 14.
   Full step-by-step solution

   Step 1: The profit function is P(x) = 9x² - 17x + 14, where x is the number of hundreds of units sold. Step 2: The constant term is 14, which does not change when x changes. Step 3: When x = 0 (no units sold), P(0) = 9(0)² - 17(0) + 14 = 14. Step 4: In a profit function, the constant term typically represents fixed costs or initial profit/loss independent of sales. Step 5: Since the constant is positive, it represents the profit when zero units are sold, such as initial capital or fixed revenue. The answer is 14.

4. Aisha is analyzing the growth of a bacteria culture in her biology lab. The population P(t) after t hours is modeled by the exponential function P(t) = 500 × 2^(t/3). She needs to determine how long it will take for the bacteria population to reach 4000. Write an equation to represent this situation and solve for t. Answer: 9 Solution: Set up the equation using the given function: 500 × 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Since the bases are equal, set the exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 Check: 500 × 2^(9/3) = 500 × 2^3 = 500 × 8 = 4000 The…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 500 × 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6: Check: 500 × 2^(9/3) = 500 × 2^3 = 500 × 8 = 4000 The answer is 9 hours.

5. The total cost C (in dollars) for Kaia's school club to produce 188 custom T-shirts is given by C = 28x + 146. In this expression, what quantity does the term 146 represent? Answer: 146 Solution: The cost function is C = 28x + 146, where x is the number of T-shirts. The term 28x represents the variable cost: $28 per shirt times x shirts. The constant term 146 does not depend on x.
   Full step-by-step solution

   Step 1: The cost function is C = 28x + 146, where x is the number of T-shirts. Step 2: The term 28x represents the variable cost: $28 per shirt times x shirts. Step 3: The constant term 146 does not depend on x. It represents the fixed costs (setup, design, equipment) that are incurred even if no shirts are produced. Step 4: Therefore, 146 represents the fixed cost of $146. Answer: $146 fixed cost.

6. Isabella's company profit is modeled by P(x) = -2x² + 27x - 7, where x represents hundreds of units sold. What quantity does the coefficient -2 represent in this context? Answer: -2 Solution: The profit function is P(x) = -2x² + 27x - 7, where x is hundreds of units sold. The leading coefficient is -2, which is attached to the x² term.
   Full step-by-step solution

   Step 1: The profit function is P(x) = -2x² + 27x - 7, where x is hundreds of units sold. Step 2: The leading coefficient is -2, which is attached to the x² term. Step 3: In a quadratic function, the leading coefficient determines the concavity of the parabola. A negative coefficient means the parabola opens downward, indicating a maximum profit point. Step 4: The magnitude 2 indicates the rate at which profit decreases per additional unit sold squared, representing diminishing returns as production increases. Step 5: Therefore, the coefficient -2 represents the rate of decrease in profit due to increasing production costs or market saturation as more units are sold. The answer is -2.

7. A right triangle is positioned on a coordinate plane with vertices at (0,0), (8,0), and (8,6). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the equation of this circumscribed circle? Answer: (x-4)^2+(y-3)^2=25 Solution: Identify that for a right triangle inscribed in a circle, the hypotenuse is the diameter. The vertices (0,0) and (8,6) form the hypotenuse.
   Full step-by-step solution

   Step 1: Identify that for a right triangle inscribed in a circle, the hypotenuse is the diameter. Step 2: The vertices (0,0) and (8,6) form the hypotenuse. Step 3: Find the center of the circle by calculating the midpoint of the hypotenuse: ((0+8)/2, (0+6)/2) = (4, 3). Step 4: Calculate the radius by finding half the length of the hypotenuse: distance between (0,0) and (8,6) is sqrt((8-0)^2 + (6-0)^2) = sqrt(64 + 36) = sqrt(100) = 10. Radius = 10/2 = 5. Step 5: Write the equation of the circle using center (4,3) and radius 5: (x-4)^2 + (y-3)^2 = 5^2. Step 6: Simplify: (x-4)^2 + (y-3)^2 = 25. The equation of the circumscribed circle is (x-4)^2 + (y-3)^2 = 25.

8. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (5,12). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the equation of this circumscribed circle? Answer: (x-2.5)²+(y-6)²=42.25 Solution: Identify the hypotenuse of the right triangle. Since the vertices are at (0,0), (5,0), and (5,12), the legs are along the x-axis and y-axis, making the hypotenuse from (0,0) to (5,12).
   Full step-by-step solution

   Step 1: Identify the hypotenuse of the right triangle. Since the vertices are at (0,0), (5,0), and (5,12), the legs are along the x-axis and y-axis, making the hypotenuse from (0,0) to (5,12). Step 2: Calculate the length of the hypotenuse using the distance formula: sqrt((5-0)² + (12-0)²) = sqrt(25 + 144) = sqrt(169) = 13. Step 3: For a right triangle, the hypotenuse is the diameter of the circumscribed circle. Therefore, the diameter is 13, and the radius is 13/2 = 6.5. Step 4: The center of the circle is the midpoint of the hypotenuse. Find the midpoint between (0,0) and (5,12): ((0+5)/2, (0+12)/2) = (2.5, 6). Step 5: Write the equation of the circle with center (2.5, 6) and radius 6.5: (x-2.5)² + (y-6)² = (6.5)². Step 6: Calculate (6.5)² = 42.25. The equation is (x-2.5)² + (y-6)² = 42.25.