Functions in Context

lessonbunny.com

Answer Key & Explanations

Functions in Context · Grade 9 · Worksheet 1

1. The function P(t) = 1200(1.08)^t models the population of Kaia's town over t years. What does P(9) represent? Answer: The population after 9 years Solution: The function P(t) = 1200(1.08)^t models population growth over time The initial population is 1200 people The growth rate is 8% per year (from the 1.08 factor) P(9) means we substitute t = 9 into the function This gives us the population after exactly 9 years Therefore, P(9) represents the…
   Full step-by-step solution

   Step 1: The function P(t) = 1200(1.08)^t models population growth over time Step 2: The initial population is 1200 people Step 3: The growth rate is 8% per year (from the 1.08 factor) Step 4: P(9) means we substitute t = 9 into the function Step 5: This gives us the population after exactly 9 years Step 6: Therefore, P(9) represents the population of Kaia's town after 9 years Answer: The population after 9 years

2. A drone is flying over a field following a parabolic path modeled by the function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters and t represents time in seconds. At what time does the drone reach its maximum height? Answer: 3 seconds Solution: We are given the height function: h(t) = -2t² + 12t + 5. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 12, c = 5.
   Full step-by-step solution

   We are given the height function: h(t) = -2t² + 12t + 5. This is a quadratic function in the form h(t) = at² + bt + c, where a = -2, b = 12, c = 5. Since the parabola opens downward (a < 0), the vertex 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 a and b. a = -2 b = 12 Step 2: Plug into the vertex formula. t = -b / (2a) t = -12 / (2 * -2) Step 3: Simplify the denominator. 2 * -2 = -4 So t = -12 / (-4) Step 4: Simplify the division. -12 / (-4) = 3 Therefore, the drone reaches its maximum height at t = 3 seconds.

3. Hana is analyzing the water depth in a tidal pool. The depth, in meters, is modeled by the function f(t) = 4sin(πt/6) + 6, where t is the time in hours after midnight. A graph of this function is drawn on a coordinate plane, with t on the horizontal axis and f(t) on the vertical axis. What is the interpretation of the maximum value of f(t) in this context? Answer: The maximum water depth in the tidal pool is 10 meters. Solution: The sine function sin(πt/6) has a maximum value of 1. Step 2: Substitute this maximum into the function: f(t) = 4(1) + 6 = 4 + 6 = 10. Step 3: Therefore, the maximum depth is 10 meters.
   Full step-by-step solution

   Step 1: The sine function sin(πt/6) has a maximum value of 1. Step 2: Substitute this maximum into the function: f(t) = 4(1) + 6 = 4 + 6 = 10. Step 3: Therefore, the maximum depth is 10 meters. The interpretation is that the highest water level in the tidal pool reaches 10 meters above the reference point. The answer is 10.

4. Liam is analyzing the profit function for his small business selling handmade candles. The profit P(x) in dollars is modeled by the quadratic function P(x) = -2x² + 80x - 600, where x represents the number of candles sold. How many candles must Liam sell to maximize his profit? Answer: 20 Solution: To find the number of candles Liam must sell to maximize profit, we use the profit function: P(x) = -2x² + 80x - 600 This is a quadratic function in the form ax² + bx + c, where: a = -2 b = 80 c = -600 Since the coefficient of x² (a = -2) is negative, the parabola opens downward.
   Full step-by-step solution

   To find the number of candles Liam must sell to maximize profit, we use the profit function: P(x) = -2x² + 80x - 600 This is a quadratic function in the form ax² + bx + c, where: a = -2 b = 80 c = -600 Since the coefficient of x² (a = -2) is negative, the parabola opens downward. This means the function has a maximum value at its vertex. For any quadratic function ax² + bx + c, the x-coordinate of the vertex (which gives the maximum or minimum point) is found using the formula: x = -b / (2a) Let's substitute our values for a and b: x = -80 / (2 × -2) First, calculate the denominator: 2 × -2 = -4 So, x = -80 / (-4) Dividing a negative by a negative gives a positive: x = 80 / 4 x = 20 Therefore, Liam must sell 20 candles to maximize his profit. Verification: At x = 20, the profit would be: P(20) = -2(20)² + 80(20) - 600 = -2(400) + 1600 - 600 = -800 + 1600 - 600 = 800 - 600 = 200 dollars This confirms that selling 20 candles gives a profit of $200, and since the parabola opens downward, this is indeed the maximum profit point. ANSWER: 20

5. The function C(x) = 27x + 72 represents the cost in dollars for Mason to produce x custom t-shirts. What does C(12) represent in this context? Answer: The total cost in dollars to produce 12 custom t-shirts Solution: The function C(x) = 27x + 72 models the total cost to produce x custom t-shirts. The fixed cost is $72 (perhaps for setup or equipment), and the variable cost per shirt is $27.
   Full step-by-step solution

   Step 1: The function C(x) = 27x + 72 models the total cost to produce x custom t-shirts. The fixed cost is $72 (perhaps for setup or equipment), and the variable cost per shirt is $27. Step 2: C(12) means we substitute x = 12 into the function. Step 3: C(12) = 27(12) + 72 = 324 + 72 = 396. Step 4: Therefore, C(12) represents the total cost in dollars to produce 12 custom t-shirts, which is $396. The answer is: The total cost in dollars to produce 12 custom t-shirts.

6. Isabella is analyzing the shape of a suspension bridge cable. The cable is modeled by the quadratic function f(x) = 0.02(x - 27)^2 + 7, where f(x) represents the height of the cable in meters above the bridge deck, and x is the horizontal distance in meters from the left tower. What is the meaning of the value 7 in this function in the context of the bridge? Answer: The lowest point of the cable is 7 meters above the bridge deck. Solution: Identify that the function is in vertex form: f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Here, a = 0.02, h = 27, and k = 7.
   Full step-by-step solution

   Step 1: Identify that the function is in vertex form: f(x) = a(x - h)^2 + k, where (h, k) is the vertex. Step 2: Here, a = 0.02, h = 27, and k = 7. Step 3: Since a = 0.02 is positive, the parabola opens upward, so the vertex is the minimum point. Step 4: The vertex is at (27, 7). This means at x = 27 meters from the left tower, the cable reaches its lowest height. Step 5: The value 7 represents the minimum height of the cable above the bridge deck. Step 6: Therefore, the meaning of 7 is that the lowest point of the cable is 7 meters above the bridge deck. The answer is: The lowest point of the cable is 7 meters above the bridge deck.

7. A drone is flying over a field following a parabolic path described by the function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters and t represents time in seconds. The drone operator wants to know the maximum height the drone reaches and at what time this occurs. What is the maximum height and when does it occur? Answer: The maximum height is 23 meters and it occurs at 3 seconds. Solution: h(t) = -2t² + 12t + 5 This is a quadratic function in the form h(t) = at² + bt + c, with a = -2, b = 12, c = 5. Since a < 0, the parabola opens downward, so the vertex is the maximum point.
   Full step-by-step solution

   Let's solve this step by step. We are given the height function: h(t) = -2t² + 12t + 5 --- **Step 1: Identify the type of function and its graph** This is a quadratic function in the form h(t) = at² + bt + c, with a = -2, b = 12, c = 5. Since a < 0, the parabola opens downward, so the vertex is the maximum point. --- **Step 2: Find the time at which maximum height occurs** For a quadratic function at² + bt + c, the vertex occurs at t = -b / (2a). t = -12 / (2 * -2) t = -12 / (-4) t = 3 seconds So the maximum height occurs at t = 3 seconds. --- **Step 3: Find the maximum height** Substitute t = 3 into h(t): h(3) = -2*(3)² + 12*3 + 5 h(3) = -2*9 + 36 + 5 h(3) = -18 + 36 + 5 h(3) = 18 + 5 h(3) = 23 meters --- **Step 4: Conclusion** The maximum height is 23 meters and it occurs at 3 seconds.

8. Sophia's smartphone battery percentage is modeled by B(t) = 85 - 12t, where t is hours of use. What does B(3) represent? Answer: 49 Solution: The function B(t) = 85 - 12t models Sophia's battery percentage after t hours of use B(3) represents the battery percentage after 3 hours of use Substitute t = 3 into the function: B(3) = 85 - 12(3) Multiply: 12 × 3 = 36 Subtract: 85 - 36 = 49 This means after 3 hours of use, Sophia's phone…
   Full step-by-step solution

   Step 1: The function B(t) = 85 - 12t models Sophia's battery percentage after t hours of use Step 2: B(3) represents the battery percentage after 3 hours of use Step 3: Substitute t = 3 into the function: B(3) = 85 - 12(3) Step 4: Multiply: 12 × 3 = 36 Step 5: Subtract: 85 - 36 = 49 Step 6: This means after 3 hours of use, Sophia's phone battery will be at 49% The answer is 49.