Interpret Expressions

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Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 2

1. Charlotte is designing a pattern for a mural. The pattern consists of a large square divided into smaller squares. The side length of the large square is (3x + 7) inches. Inside it, there is a shaded square whose side length is (x + 2) inches. The expression representing the area of the unshaded region (the part of the large square not covered by the shaded square) is (3x + 7)^2 - (x + 2)^2. What quantity does the term (x + 2)^2 represent in the context of this mural? Answer: The area of the shaded square in square inches. Solution: Identify the context. The large square has side length (3x + 7) inches, and the shaded square has side length (x + 2) inches.
   Full step-by-step solution

   Step 1: Identify the context. The large square has side length (3x + 7) inches, and the shaded square has side length (x + 2) inches. Step 2: The expression (3x + 7)^2 represents the area of the large square because area = side length squared. Step 3: Similarly, (x + 2)^2 represents the area of the shaded square because its side length is (x + 2) inches. Step 4: Therefore, the quantity (x + 2)^2 is the area of the shaded square in square inches. The answer is: The area of the shaded square in square inches.

2. Noah's rocket height is modeled by h(t) = -4.9t² + 49t + 2.5, where t is time in seconds. What quantity does the constant term 2.5 represent in this context? Answer: 2.5 Solution: The height function is h(t) = -4.9t² + 49t + 2.5, where t represents time in seconds after launch. The constant term is 2.5, which does not depend on t. Evaluate h(0) = -4.9(0)² + 49(0) + 2.5 = 0 + 0 + 2.5 = 2.5.
   Full step-by-step solution

   Step 1: The height function is h(t) = -4.9t² + 49t + 2.5, where t represents time in seconds after launch. Step 2: The constant term is 2.5, which does not depend on t. Step 3: Evaluate h(0) = -4.9(0)² + 49(0) + 2.5 = 0 + 0 + 2.5 = 2.5. Step 4: At t = 0 seconds, the rocket has not yet launched, so h(0) represents the initial height above ground. Step 5: Therefore, the constant term 2.5 represents the initial height of the rocket in meters above the ground before launch. The answer is 2.5.

3. Mere's smartphone data plan charges a base fee of $12 plus $0.04 per megabyte used. The expression 12 + 0.04m represents the total monthly cost. What quantity does the number 0.04 represent in this context? Answer: 0.04 Solution: The expression is 12 + 0.04m, where m represents megabytes used. The number 12 is the fixed base fee that doesn't change. The term 0.04m represents the variable cost that depends on data usage.
   Full step-by-step solution

   Step 1: The expression is 12 + 0.04m, where m represents megabytes used. Step 2: The number 12 is the fixed base fee that doesn't change. Step 3: The term 0.04m represents the variable cost that depends on data usage. Step 4: The coefficient 0.04 indicates the cost per unit of data used. Step 5: Therefore, 0.04 represents the cost per megabyte of data used. The answer is 0.04.

4. Isabella's company profit is modeled by P(x) = 12x² - 27x + 7 where x represents hundreds of units sold. What does the constant term 7 represent in this context? Answer: 7 Solution: In the profit function P(x) = 12x² - 27x + 7, the variable x represents hundreds of units sold. Step 2: The constant term 7 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 - 27x + 7, the variable x represents hundreds of units sold. Step 2: The constant term 7 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. 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 7 represents the profit (in appropriate monetary units) when zero units are sold. The answer is 7.

5. 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.

6. 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.

7. Sophia's company profit is modeled by P(x) = -3x² + 48x - 15, where x represents hundreds of units sold and P(x) is profit in thousands of dollars. What quantity does the constant term -15 represent in this context? Answer: the profit (in thousands of dollars) when zero units are sold Solution: The profit function is P(x) = -3x² + 48x - 15, where x is hundreds of units sold and P(x) is profit in thousands of dollars. The constant term is -15, which does not depend on x.
   Full step-by-step solution

   Step 1: The profit function is P(x) = -3x² + 48x - 15, where x is hundreds of units sold and P(x) is profit in thousands of dollars. Step 2: The constant term is -15, which does not depend on x. Step 3: When x = 0 (no units sold), P(0) = -3(0)² + 48(0) - 15 = 0 + 0 - 15 = -15. Step 4: This means the profit when no units are sold is -15 thousand dollars, or a loss of $15,000. Step 5: In business, this often represents fixed costs or initial profit/loss. Since it is negative, it represents fixed costs or initial losses before any sales occur. The answer is: the profit (in thousands of dollars) when zero units are sold.

8. 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.