Mathematics: Applications & Interpretation SL
Functions and modelling — Topic 2
- 1.
A gym charges a joining fee plus a fixed cost per visit. A member who makes 2 visits pays a total of $15, and a member who makes 6 visits pays a total of $35. Assuming the relationship is linear, find the equation for the total cost C in terms of the number of visits n, and interpret the gradient and the C-intercept in context.
[4 marks] - 2.
Marking analysis: A learner attempts the following task: “A gym charges a joining fee plus a fixed cost per visit. A member who makes 2 visits pays a total of $15, and a member who makes 6 visits pays a total of $35. Assuming the relationship is linear, find the equation for the total cost C in terms of the number of visits n, and interpret the gradient and the C-intercept in context.” Their response addresses only this point: “Calculates the gradient as (35 − 15)/(6 − 2) = 5.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 3.
The height of a ball thrown in the air is modelled by h(t) = −5t² + 20t + 1, where h is in metres and t is in seconds. (a) Find the time at which the ball reaches its maximum height. (b) Find the maximum height reached.
[3 marks] - 4.
Marking analysis: A learner attempts the following task: “The height of a ball thrown in the air is modelled by h(t) = −5t² + 20t + 1, where h is in metres and t is in seconds. (a) Find the time at which the ball reaches its maximum height. (b) Find the maximum height reached.” Their response addresses only this point: “Uses the vertex formula t = −b/(2a) with a = −5, b = 20 to obtain t = 2.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 5.
A bacteria population grows exponentially according to P(t) = P0·e^(kt). The initial population is 200, and after 5 hours the population is 600. (a) Find the value of k. (b) Use your model to predict the population after 10 hours.
[4 marks] - 6.
Marking analysis: A learner attempts the following task: “A bacteria population grows exponentially according to P(t) = P0·e^(kt). The initial population is 200, and after 5 hours the population is 600. (a) Find the value of k. (b) Use your model to predict the population after 10 hours.” Their response addresses only this point: “Substitutes P(5) = 600 and P0 = 200 into 600 = 200e^(5k).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 7.
A radioactive substance has a half-life of 8 days. A sample initially has a mass of 50 grams. (a) Find the decay constant k in the model m(t) = 50e^(−kt). (b) Find the mass remaining after 20 days.
[4 marks] - 8.
Marking analysis: A learner attempts the following task: “A radioactive substance has a half-life of 8 days. A sample initially has a mass of 50 grams. (a) Find the decay constant k in the model m(t) = 50e^(−kt). (b) Find the mass remaining after 20 days.” Their response addresses only this point: “Uses the half-life condition m(8) = 25 to set up 25 = 50e^(−8k).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 9.
The sound intensity level L in decibels is modelled by L = 10 log10(I/I0), where I is the intensity in W/m² and I0 = 1 × 10⁻¹² W/m² is the reference intensity. Calculate the sound intensity level of a sound with intensity I = 1 × 10⁻⁶ W/m².
[3 marks] - 10.
Marking analysis: A learner attempts the following task: “The sound intensity level L in decibels is modelled by L = 10 log10(I/I0), where I is the intensity in W/m² and I0 = 1 × 10⁻¹² W/m² is the reference intensity. Calculate the sound intensity level of a sound with intensity I = 1 × 10⁻⁶ W/m².” Their response addresses only this point: “Substitutes I = 1×10⁻⁶ and I0 = 1×10⁻¹² into the formula.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 11.
The cost y of buying x identical notebooks varies directly with x. If 6 notebooks cost $24, find the constant of variation, and use it to find the cost of 10 notebooks.
[3 marks] - 12.
Marking analysis: A learner attempts the following task: “The cost y of buying x identical notebooks varies directly with x. If 6 notebooks cost $24, find the constant of variation, and use it to find the cost of 10 notebooks.” Their response addresses only this point: “Sets up y = kx and substitutes y = 24, x = 6 to obtain k = 4.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 13.
The time y taken to complete a job varies inversely with the number of workers x. If 8 workers take 5 hours, find the constant of variation, and use it to find how long 20 workers would take.
[3 marks] - 14.
Marking analysis: A learner attempts the following task: “The time y taken to complete a job varies inversely with the number of workers x. If 8 workers take 5 hours, find the constant of variation, and use it to find how long 20 workers would take.” Their response addresses only this point: “Sets up y = k/x and substitutes y = 5, x = 8 to obtain k = 40.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 15.
A parking garage charges according to the piecewise function C(h) = 5 if 0 < h ≤ 1, C(h) = 5 + 3(h − 1) if h > 1, where h is the number of hours parked and C is the cost in dollars. Calculate the cost of parking for (a) 1 hour, and (b) 4 hours.
[3 marks] - 16.
Marking analysis: A learner attempts the following task: “A parking garage charges according to the piecewise function C(h) = 5 if 0 < h ≤ 1, C(h) = 5 + 3(h − 1) if h > 1, where h is the number of hours parked and C is the cost in dollars. Calculate the cost of parking for (a) 1 hour, and (b) 4 hours.” Their response addresses only this point: “Identifies that h = 1 falls in the first piece, giving a cost of $5.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 17.
Solve the equation 3e^(2x) = 45 for x, giving your answer to 3 significant figures.
[3 marks] - 18.
Marking analysis: A learner attempts the following task: “Solve the equation 3e^(2x) = 45 for x, giving your answer to 3 significant figures.” Their response addresses only this point: “Divides both sides by 3 to obtain e^(2x) = 15.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 19.
Solve the equation log2(x) + log2(x − 2) = 3 for x.
[4 marks] - 20.
Marking analysis: A learner attempts the following task: “Solve the equation log2(x) + log2(x − 2) = 3 for x.” Their response addresses only this point: “Uses the law of logarithms to combine the left side: log2[x(x − 2)] = 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 21.
Given f(x) = 2x + 1 and g(x) = x² − 3, find (a) f(g(2)) and (b) g(f(2)).
[3 marks] · no calculator - 22.
Marking analysis: A learner attempts the following task: “Given f(x) = 2x + 1 and g(x) = x² − 3, find (a) f(g(2)) and (b) g(f(2)).” Their response addresses only this point: “Calculates g(2) = 2² − 3 = 1, then f(1) = 2(1) + 1 = 3, giving f(g(2)) = 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 23.
Find the inverse function f⁻¹(x) for f(x) = (3x − 2)/5, and verify your answer by finding f⁻¹(f(4)).
[4 marks] · no calculator - 24.
Marking analysis: A learner attempts the following task: “Find the inverse function f⁻¹(x) for f(x) = (3x − 2)/5, and verify your answer by finding f⁻¹(f(4)).” Their response addresses only this point: “Sets y = (3x − 2)/5 and swaps x and y to begin solving for the inverse.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 25.
A model predicts the number of visitors to a museum as V(x) = 100√(20 − x), where x is the entry price in dollars. Explain why the domain of this model should be restricted to 0 ≤ x ≤ 20 in this real-world context.
[2 marks] · no calculator - 26.
Marking analysis: A learner attempts the following task: “A model predicts the number of visitors to a museum as V(x) = 100√(20 − x), where x is the entry price in dollars. Explain why the domain of this model should be restricted to 0 ≤ x ≤ 20 in this real-world context.” Their response addresses only this point: “Explains that x cannot be negative, since an entry price cannot be below zero, giving the lower bound x ≥ 0.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator - 27.
The graph of y = x² is transformed to the graph of y = (x − 3)² + 2. Describe the two transformations applied, and state the coordinates of the vertex of the transformed graph.
[2 marks] · no calculator - 28.
Marking analysis: A learner attempts the following task: “The graph of y = x² is transformed to the graph of y = (x − 3)² + 2. Describe the two transformations applied, and state the coordinates of the vertex of the transformed graph.” Their response addresses only this point: “Describes a horizontal translation of 3 units to the right, and a vertical translation of 2 units up.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator - 29.
The height of the tide in a harbour, h metres, t hours after midnight, is modelled by h(t) = 5 + 3sin(πt/6). (a) State the amplitude and the period of this model. (b) Calculate the height of the tide at t = 3 hours.
[4 marks] - 30.
Marking analysis: A learner attempts the following task: “The height of the tide in a harbour, h metres, t hours after midnight, is modelled by h(t) = 5 + 3sin(πt/6). (a) State the amplitude and the period of this model. (b) Calculate the height of the tide at t = 3 hours.” Their response addresses only this point: “States the amplitude as 3 metres.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 31.
Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x² − 4x + 5.
[5 marks] - 32.
Marking analysis: A learner attempts the following task: “Find the coordinates of the points of intersection of the line y = 2x + 1 and the curve y = x² − 4x + 5.” Their response addresses only this point: “Sets the two expressions for y equal: 2x + 1 = x² − 4x + 5.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]