Interpret Expressions

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Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 2

1. Isabella's company profit is modeled by P(x) = 7x² - 12x + 17 where x represents hundreds of units sold. What does the constant term 17 represent in this context? Answer: 17 Solution: In the profit function P(x) = 7x² - 12x + 17, the variable x represents hundreds of units sold. The constant term 17 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) = 7x² - 12x + 17, the variable x represents hundreds of units sold. Step 2: The constant term 17 is independent of x, meaning it does not change based on the number of units sold. Step 3: In business profit functions, constant terms typically represent fixed costs or initial investments that must be paid regardless of sales, or initial capital. Step 4: Since this is a profit function and the constant is positive, it likely represents initial capital or fixed revenue sources that exist even when no units are sold. Step 5: The constant term 17 represents the profit (in appropriate monetary units) when zero units are sold. The answer is 17.

2. Emma is analyzing the growth of a bacteria culture in her biology lab. The population P after t hours is modeled by the 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 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.

3. Noah's company profit is modeled by P(x) = 9x² - 17x + 13 where x represents hundreds of units sold. What does the constant term 13 represent in this context? Answer: 13 Solution: In the profit function P(x) = 9x² - 17x + 13, the variable x represents hundreds of units sold. The constant term 13 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) = 9x² - 17x + 13, the variable x represents hundreds of units sold. Step 2: The constant term 13 is independent of x, meaning it does not change based on the number of units sold. Step 3: To find what it represents, evaluate P(0): P(0) = 9(0)² - 17(0) + 13 = 13. Step 4: This means when zero units are sold, the profit is 13 (in appropriate monetary units). Step 5: In business profit functions, constant terms typically represent fixed costs or initial investments. Since the constant is positive, it likely represents initial capital or fixed revenue sources. The answer is 13.

4. Charlotte's rocket height is modeled by h(t) = -5t² + 40t + 15 where t represents time in seconds. What quantity does the constant term 15 represent in this context? Answer: 15 Solution: The function h(t) = -5t² + 40t + 15 gives the height of the rocket at time t seconds. The constant term is the part of the expression that does not depend on t, which is 15.
   Full step-by-step solution

   Step 1: The function h(t) = -5t² + 40t + 15 gives the height of the rocket at time t seconds. Step 2: The constant term is the part of the expression that does not depend on t, which is 15. Step 3: To understand what it represents, evaluate h(0): h(0) = -5(0)² + 40(0) + 15 = 15. Step 4: This means at t = 0 seconds (the moment of launch), the rocket's height is 15 units. Step 5: Therefore, the constant term 15 represents the initial height of the rocket above the ground at launch time. The answer is 15.

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). 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)² + (y - 1.5)² = 6.25 Solution: Identify the triangle's vertices and properties. Vertices: A = (0,0), B = (4,0), C = (4,3).
   Full step-by-step solution

   Step 1: Identify the triangle's vertices and properties. Vertices: A = (0,0), B = (4,0), C = (4,3). Since B and C share the same x-coordinate (x=4), and A and B share the same y-coordinate (y=0), AB is horizontal and BC is vertical. Therefore, angle B is 90°, so the triangle is right-angled at B(4,0). Step 2: In a right triangle, the circumcenter (center of the circumscribed circle) is the midpoint of the hypotenuse. The hypotenuse is AC (opposite the right angle). Find A = (0,0) and C = (4,3). Midpoint M = ((0+4)/2, (0+3)/2) = (4/2, 3/2) = (2, 1.5). So the center O = (2, 1.5). Step 3: Find the radius. The radius is the distance from O to any vertex, e.g., A(0,0). Distance OA = sqrt((2-0)^2 + (1.5-0)^2) = sqrt(4 + 2.25) = sqrt(6.25) = 2.5. Step 4: Write the equation of the circle. Center (h,k) = (2, 1.5), radius r = 2.5. Equation: (x - 2)^2 + (y - 1.5)^2 = (2.5)^2. (2.5)^2 = 6.25. Final equation: (x - 2)^2 + (y - 1.5)^2 = 6.25.

6. Mere's rectangular garden has length 12 meters and width 8 meters. The expression 2(12 + 8) represents what quantity? Answer: 40 Solution: The expression is 2(12 + 8), where 12 represents the length and 8 represents the width First, add the length and width: 12 + 8 = 20 Then multiply by 2: 2 × 20 = 40 This calculation gives us the perimeter of the rectangle The perimeter represents the total distance around Mere's garden The answer…
   Full step-by-step solution

   Step 1: The expression is 2(12 + 8), where 12 represents the length and 8 represents the width Step 2: First, add the length and width: 12 + 8 = 20 Step 3: Then multiply by 2: 2 × 20 = 40 Step 4: This calculation gives us the perimeter of the rectangle Step 5: The perimeter represents the total distance around Mere's garden The answer is 40 meters.

7. The total cost C (in dollars) for Lucas's school club to produce 70 custom T-shirts is given by C = 28x + 32. In this expression, what quantity does the term 32 represent? Answer: 32 Solution: The cost function is C = 28x + 32, where x is the number of T-shirts. The term 28x represents the variable cost: $28 per shirt times x shirts. The constant term 32 does not depend on x.
   Full step-by-step solution

   Step 1: The cost function is C = 28x + 32, 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 32 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, 32 represents the fixed cost of $32. Answer: $32 fixed cost.