Mathematical Models
If you're planning a road trip, list three variables you might consider when calculating the total cost of the trip.
Any three variables such as distance, fuel efficiency, fuel cost, food cost, lodging cost, etc.
Solve for x:
2x = 10
x = 10/2 = 5
Solve these simultaneous equations:
y = 2x and y = x + 3
(3,6)
What is the area of a circle with a radius of 3 units?
9π square units
If two variables x and y are directly proportional and x=10 when y=5, what is x when y=15?
30
How many variables are needed to calculate the area of a rectangle?
2
Rearrange the formula F = C * 9/5 + 32 to solve for C.
C = (F - 32) * 5/9
Solve these simultaneous equations:
3x + 2y = 11 and 6x + 4y = 40
trick question: there is no solution!
If a circle has a diameter of 8 units, what is the radius?
4 units
If x and y are inversely proportional and x=4 when y=2, what is x when y=8?
1
Write a mathematical model for the monthly cost (C) of a gym membership if there's a joining fee (J) and a monthly fee (M).
C = J + M
Solve for y, given x = 2:
3x + 2y = 12
y = 6/2 = 3
Solve these simultaneous equations:
2x + 3y = 11 and 4x - 3y = 8
(19/6 , 13/9)
Calculate the area of a sector that is 1/4 of a circle with a radius of 4 units.
4π square units
If a task takes 3 man-hours to complete and 2 people are working on it, how long will it take to complete the task?
1.5 hours
Give examples of mathematical models that use 1, 2, and 3 variables, respectively.
(Hint: think geometric area formulas)
1 variable: square, cube, etc.
2 variables: rectangle, triangle, etc.
3 variables: trapezoid, cuboid, etc.
Solve for x:
4x2 = 16
x = ±2
Solve these simultaneous equations:
x + 2y = 7 and 2x + y = 8
(3,2)
A bicycle wheel has a diameter of 0.7 meters. If the wheel completes 2000 rotations, how far (in kilometers) has the bicycle traveled?
1.4π kilometers
If 4 people can paint a house in 5 hours, how long would it take 10 people to paint the same house?
2 hours
In a mathematical model of a car's fuel efficiency, what other variables might you add beyond just distance travelled and fuel consumed?
Additional factors could include the car's speed, tire inflation, car's weight, engine efficiency, type of fuel, weather conditions, etc.
Solve for x:
5x^2 + 3x - 2 = 0
x = -1
Solve these simultaneous equations:
y = 2x + 1 and y = -x + 3
(2/3, 5/3)
Consider a sector of a circle with a radius of 6 units and a central angle of 60 degrees. What is the perimeter of this sector?
π + 12 units
If 4 people can complete a task in 3 hours, how many hours are saved by adding a fifth person, assuming the work rate remains constant?
0.6 hours (or 36 minutes)