Functions in Context

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Functions in Context · Grade 9 · Worksheet 3

1. The height of a plant is modeled by h(t) = 15 + 0.8t, where t is days and h is cm. What does h(12) represent? Answer: 24.6 Solution: The function h(t) = 15 + 0.8t models plant height h(12) means the height after 12 days Substitute t = 12 into the function: h(12) = 15 + 0.8 × 12 Calculate multiplication: 0.8 × 12 = 9.6 Add the initial height: 15 + 9.6 = 24.6 This means after 12 days, the plant will be 24.6 cm tall The answer…
   Full step-by-step solution

   Step 1: The function h(t) = 15 + 0.8t models plant height Step 2: h(12) means the height after 12 days Step 3: Substitute t = 12 into the function: h(12) = 15 + 0.8 × 12 Step 4: Calculate multiplication: 0.8 × 12 = 9.6 Step 5: Add the initial height: 15 + 9.6 = 24.6 Step 6: This means after 12 days, the plant will be 24.6 cm tall The answer is 24.6.

2. The population of a certain species of birds in a wildlife reserve is modeled by the function P(t) = 1200/(1 + 3e^(-0.2t)), where t is the number of years since monitoring began and P(t) is the population size. According to this model, what is the maximum carrying capacity (in number of birds) that the reserve can support? Answer: 1200 Solution: The population function is given as P(t) = 1200/(1 + 3e^(-0.2t)) This is a logistic growth model of the form P(t) = L/(1 + Ce^(-kt)), where L represents the carrying capacity As t approaches infinity, e^(-0.2t) approaches 0 When e^(-0.2t) approaches 0, the denominator (1 + 3e^(-0.2t)) approaches…
   Full step-by-step solution

   Step 1: The population function is given as P(t) = 1200/(1 + 3e^(-0.2t)) Step 2: This is a logistic growth model of the form P(t) = L/(1 + Ce^(-kt)), where L represents the carrying capacity Step 3: As t approaches infinity, e^(-0.2t) approaches 0 Step 4: When e^(-0.2t) approaches 0, the denominator (1 + 3e^(-0.2t)) approaches 1 Step 5: Therefore, P(t) approaches 1200/1 = 1200 as t approaches infinity Step 6: The carrying capacity is the maximum population the environment can sustain, which is 1200 birds The answer is 1200.

3. Aroha is tracking the temperature of a chemical reaction over time. The temperature T(t) in degrees Celsius after t minutes is modeled by the quadratic function T(t) = -2t² + 24t + 10. Interpret the meaning of T(6) in the context of this reaction and determine its value. Answer: T(6) = 82, meaning after 6 minutes, the temperature of the chemical reaction is 82 degrees Celsius. Solution: T(t) = -2t² + 24t + 10 models the temperature in degrees Celsius after t minutes. To find T(6), substitute t = 6 into the function. T(6) = -2(6)² + 24(6) + 10 Calculate (6)² = 36.
   Full step-by-step solution

   Step 1: Understand the function. T(t) = -2t² + 24t + 10 models the temperature in degrees Celsius after t minutes. Step 2: To find T(6), substitute t = 6 into the function. Step 3: T(6) = -2(6)² + 24(6) + 10 Step 4: Calculate (6)² = 36. Step 5: Multiply: -2 × 36 = -72. Step 6: Multiply: 24 × 6 = 144. Step 7: Add the terms: -72 + 144 + 10 = 82. Step 8: T(6) = 82, meaning after 6 minutes, the temperature of the chemical reaction is 82 degrees Celsius. The answer is T(6) = 82, interpreted as the temperature after 6 minutes.

4. A right circular cone has a height of 12 cm and a base radius of 5 cm. A horizontal cross-section is taken 4 cm from the vertex, creating a smaller cone at the top and a frustum below. What is the radius of this circular cross-section? Answer: 1.67 Solution: - Height \( h = 12 \) cm - Base radius \( R = 5 \) cm A horizontal cross-section is taken 4 cm from the vertex. That means from the tip (vertex) down, at distance 4 cm, we cut horizontally.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a right circular cone: - Height \( h = 12 \) cm - Base radius \( R = 5 \) cm A horizontal cross-section is taken 4 cm from the vertex. That means from the tip (vertex) down, at distance 4 cm, we cut horizontally. This creates a small cone on top (height 4 cm) and a frustum below. We need the radius \( r \) of the cross-section circle at that height. --- **Step 2: Use similar triangles** The whole cone has height 12 and base radius 5. If we take a smaller cone of height 4 (from vertex to cross-section), it is similar to the whole cone. In similar cones, the ratio of corresponding lengths is constant: \[ \frac{\text{radius of small cone}}{\text{height of small cone}} = \frac{\text{radius of big cone}}{\text{height of big cone}} \] That is: \[ \frac{r}{4} = \frac{5}{12} \] --- **Step 3: Solve for \( r \)** Multiply both sides by 4: \[ r = 4 \times \frac{5}{12} \] \[ r = \frac{20}{12} \] \[ r = \frac{5}{3} \] --- **Step 4: Convert to decimal** \[ \frac{5}{3} = 1.666\ldots \] Rounded to two decimal places: \( 1.67 \) --- **Final Answer:** 1.67

5. Charlotte is a civil engineer designing a suspension bridge. The height of the main cable above the bridge deck, in meters, is modeled by the quadratic function h(x) = 0.01x^2 - 1.2x + 48, where x is the horizontal distance in meters from the left tower. What is the minimum height of the cable above the deck, and at what horizontal distance from the left tower does this minimum occur? Answer: The minimum height is 12 meters, occurring 60 meters from the left tower. Solution: Identify the coefficients in h(x) = 0.01x^2 - 1.2x + 48. Here, a = 0.01, b = -1.2, c = 48. The x-coordinate of the vertex (axis of symmetry) is given by x = -b / (2a).
   Full step-by-step solution

   Step 1: Identify the coefficients in h(x) = 0.01x^2 - 1.2x + 48. Here, a = 0.01, b = -1.2, c = 48. Step 2: The x-coordinate of the vertex (axis of symmetry) is given by x = -b / (2a). Step 3: Substitute the values: x = -(-1.2) / (2 * 0.01) = 1.2 / 0.02 = 60. Step 4: So the minimum height occurs at x = 60 meters from the left tower. Step 5: Find the minimum height by substituting x = 60 into the function: h(60) = 0.01 * (60)^2 - 1.2 * 60 + 48. Step 6: Calculate 60^2 = 3600. Then 0.01 * 3600 = 36. Next, 1.2 * 60 = 72. So h(60) = 36 - 72 + 48. Step 7: Simplify: 36 - 72 = -36, and -36 + 48 = 12. Step 8: The minimum height is 12 meters. The answer is: The minimum height is 12 meters, occurring 60 meters from the left tower.

6. Mason is tracking the height of a sunflower he planted. The height of the sunflower, h(d), in centimeters, after d days since it sprouted is modeled by the function h(d) = 2.7d + 12. What does h(17) represent in this context, and what is its value? Answer: The height of the sunflower after 17 days is 57.9 centimeters. Solution: Identify the variables. d represents the number of days since the sunflower sprouted. To find h(17), substitute d = 17 into the function: h(17) = 2.7(17) + 12.
   Full step-by-step solution

   Step 1: Identify the variables. d represents the number of days since the sunflower sprouted. h(d) represents the height in centimeters after d days. Step 2: To find h(17), substitute d = 17 into the function: h(17) = 2.7(17) + 12. Step 3: Multiply 2.7 by 17: 2.7 * 17 = 45.9. Step 4: Add 12: 45.9 + 12 = 57.9. Step 5: Interpret: h(17) = 57.9 means that after 17 days, the sunflower is 57.9 centimeters tall. The answer is: The height of the sunflower after 17 days is 57.9 centimeters.