Practice SL 2.1—Equations of a line with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A manufacturing company observes that its profit increases by $500 for every 50 units sold. Assume that the profit is directly proportional to the number of units sold.
Find the rate of change of the profit function.
If the company sells 200 units, calculate the total profit.
If the goal is to reach a profit of $3,000, determine how many units the company must sell.
In urban planning, architects must ensure that roofs have a sufficient slope to facilitate rainwater drainage. For effective drainage, a roof must have a slope of at least degrees.
Express the slope as a ratio of vertical rise to horizontal run in the form .
If the horizontal width of the roof is meters, calculate the vertical height difference between the two sides to meet the -degree requirement.
A new eco-friendly transportation route is being constructed parallel to an existing highway to reduce traffic congestion. The highway follows the equation .
Find the equations of the possible new transportation routes that run parallel to the highway and maintain a distance of from it.
Write the equation of one of the new routes in intercept form.
Write the equation of the same route in normal form.
A new park is being designed, and the planners need to place a fountain at a specific point relative to the line segment connecting two landmarks, and .
Find the coordinates of point on the line segment such that it is twice as far from as it is from .
Find the coordinates of point on the line passing through and , such that it is twice as far from as it is from , and lies between and .
An electric airplane is making a sustainable descent towards an airport. The airplane starts its descent at a height of metres, located kilometres away from the airport, and lands directly at the airport. Let be the horizontal distance from the starting point in kilometres and be the height of the airplane in metres.
Find the equation of the line that models the airplane's descent path, assuming a constant descent angle.
How high will the airplane be when it is kilometres from the airport?
Consider the two lines and .
Find the acute angle between lines and .
Determine if the system of equations and is consistent.
Find the distance between the two given lines if they are parallel.
Sketch the lines and .
A ski resort is evaluating its beginner slope, which descends 120 metres over a horizontal distance of 500 metres.
Calculate the slope of the ski run. Give your answer as a decimal.
If a skier starts at the top of the slope and skis 200 metres horizontally, how much elevation will they lose?
If the resort classifies slopes with a gradient greater than 30% as intermediate, would this slope be classified as beginner or intermediate? Show your working.
A ramp is being designed to provide wheelchair access to a public building. The ramp must comply with the standard that requires a maximum slope of 1:12.
Calculate the slope of the ramp if it needs to rise 1 meter over a horizontal distance of 15 meters.
Determine whether this ramp meets the maximum slope requirement.
If the total rise needed is 1.5 meters, calculate the minimum horizontal distance required to meet the slope standard.
If the available horizontal space for the ramp is only 12 meters, determine the maximum rise that this ramp can achieve while meeting the slope standard.
Two cities, and , are connected by a straight road.
Find the equation of the road passing through these points in slope-intercept form.
Write the equation of the road in standard form.
Write the equation of the road in point-slope form, using the midpoint of and as a point on the line.
A solar energy company is planning to install solar panels on the roof of a community centre. The roofline can be modelled by the equation . To maximize sunlight exposure, the solar panels must be installed along a line that is perpendicular to the roofline.
Find the equation of the line along which the solar panels should be installed, passing through the point and perpendicular to the roofline. Give your answer in the form .
Write the equation of the solar panel line in the form , where .
Practice SL 2.1—Equations of a line with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A manufacturing company observes that its profit increases by $500 for every 50 units sold. Assume that the profit is directly proportional to the number of units sold.
Find the rate of change of the profit function.
If the company sells 200 units, calculate the total profit.
If the goal is to reach a profit of $3,000, determine how many units the company must sell.
In urban planning, architects must ensure that roofs have a sufficient slope to facilitate rainwater drainage. For effective drainage, a roof must have a slope of at least degrees.
Express the slope as a ratio of vertical rise to horizontal run in the form .
If the horizontal width of the roof is meters, calculate the vertical height difference between the two sides to meet the -degree requirement.
A new eco-friendly transportation route is being constructed parallel to an existing highway to reduce traffic congestion. The highway follows the equation .
Find the equations of the possible new transportation routes that run parallel to the highway and maintain a distance of from it.
Write the equation of one of the new routes in intercept form.
Write the equation of the same route in normal form.
A new park is being designed, and the planners need to place a fountain at a specific point relative to the line segment connecting two landmarks, and .
Find the coordinates of point on the line segment such that it is twice as far from as it is from .
Find the coordinates of point on the line passing through and , such that it is twice as far from as it is from , and lies between and .
An electric airplane is making a sustainable descent towards an airport. The airplane starts its descent at a height of metres, located kilometres away from the airport, and lands directly at the airport. Let be the horizontal distance from the starting point in kilometres and be the height of the airplane in metres.
Find the equation of the line that models the airplane's descent path, assuming a constant descent angle.
How high will the airplane be when it is kilometres from the airport?
Consider the two lines and .
Find the acute angle between lines and .
Determine if the system of equations and is consistent.
Find the distance between the two given lines if they are parallel.
Sketch the lines and .
A ski resort is evaluating its beginner slope, which descends 120 metres over a horizontal distance of 500 metres.
Calculate the slope of the ski run. Give your answer as a decimal.
If a skier starts at the top of the slope and skis 200 metres horizontally, how much elevation will they lose?
If the resort classifies slopes with a gradient greater than 30% as intermediate, would this slope be classified as beginner or intermediate? Show your working.
A ramp is being designed to provide wheelchair access to a public building. The ramp must comply with the standard that requires a maximum slope of 1:12.
Calculate the slope of the ramp if it needs to rise 1 meter over a horizontal distance of 15 meters.
Determine whether this ramp meets the maximum slope requirement.
If the total rise needed is 1.5 meters, calculate the minimum horizontal distance required to meet the slope standard.
If the available horizontal space for the ramp is only 12 meters, determine the maximum rise that this ramp can achieve while meeting the slope standard.
Two cities, and , are connected by a straight road.
Find the equation of the road passing through these points in slope-intercept form.
Write the equation of the road in standard form.
Write the equation of the road in point-slope form, using the midpoint of and as a point on the line.
A solar energy company is planning to install solar panels on the roof of a community centre. The roofline can be modelled by the equation . To maximize sunlight exposure, the solar panels must be installed along a line that is perpendicular to the roofline.
Find the equation of the line along which the solar panels should be installed, passing through the point and perpendicular to the roofline. Give your answer in the form .
Write the equation of the solar panel line in the form , where .