A tablet manufacturer graphed the revenue and the cost of producing each new tablet:
Calculate the selling price for each tablet.
Determine the manufacturer's fixed costs.
How many tablets does the manufacturer need to sell to cover the cost of making the tablets?
Tracy sells cookies for \$4.70 each. It costs \$2.50 to make each cookie and \$880 for the equipment needed to make the cookies.
Let x be the number of cookies sold.
Write an expression for the income function, I, in terms of x.
Write an expression for the cost function, C, in terms of x.
Sketch the graphs of the two functions on the same coordinate plane.
Solve for x, the number of units sold at which Tracy will break even.
Find the range of values of x which will Tracy turn a profit.
Find the range of values of x which will Tracy turn a loss.
What does the gradient of the income function represent?
What does the gradient of the cost function represent?
What does the y -intercept of the cost function represent?
Given the cost function C \left( x \right) = 0.3 x + 1275 and the revenue function R \left( x \right) = 2 x, find the number of units x that must be sold to break even.
The cost for a furniture manufacturer to make a dining table is \$450 per dining table plus a fixed setup cost of \$6000. The dining tables will sell for \$700 each.
Calculate the cost of manufacturing 42 dining tables.
Write an expression to represent the cost of manufacturing x dining tables.
Find the revenue that is generated by the sale of 42 dining tables.
Write an expression to represent the revenue generated from the sale of x dining tables.
If n is the whole number of dining tables that need to be sold for the company to break even, find n.
The two equations y = 3 x + 35 and y = 4 x represent Laura’s living expenses and income from work respectively.
Determine the values of x and y which satisfy both equations.
Explain the meaning of the point of intersection in this context.
The following graph shows two lines that represent the revenue and cost from selling sandwiches at a local fair:
State the coordinates of the break even point.
Find the equation of the line representing the revenue, y, from the sale of x sandwiches.
Find the equation of the line representing the cost, y, of making x sandwiches.
Would the stand make a profit or a loss if they sold 70 sandwiches?
The cost for a furniture manufacturer to make an armchair is \$600 per armchair plus a fixed setup cost of \$8500. The armchairs will sell for \$850 each.
Write an expression to represent the cost of manufacturing x armchairs.
Write an expression to represent the revenue generated from the sale of x armchairs.
Find the break-even point.
Explain the meaning of the break-even point.
A band plans to record a demo at a local studio. The cost of renting Studio A is \$250 plus \$50 per hour. The cost of renting Studio B is \$50 plus \$100 per hour.
The cost, y, in dollars of renting the studios for x hours can be modelled by the following linear system:
Studio A: y = 50 x + 250
Studio B: y = 100 x + 50
Graph the lines of both equations on the same graph.
State the values of x and y which satisfy both equations.
Interpret the solution of the equations in context.
Consider the following phone plans:
GO SMALL plan: This plan has a \$20 monthly base charge and charges 90 cents per minute for all calls.
GO MEDIUM plan: This plan has a \$26 monthly base charge and then charges 70 cents per minute for all calls.
Complete the following table of values for various total monthly call times for the two plans:
| \text{Call time }\\\ \text{(in minutes)} | \text{Total cost for}\\ \text{GO SMALL plan} | \text{Total cost for} \\ \text{GO MEDIUM plan} |
|---|---|---|
| 20 | ||
| 30 | ||
| 40 | ||
| 50 |
Write the equations for the two plans where x is the call time in minutes and y is the monthly cost in dollars.
Sketch the graphs of the two plans on the same coordinate plane.
Hence determine how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.
The sum of two numbers is 56 and their difference is 30.
Set up two equations by letting x and y be the two numbers.
Solve the system of equations.
Consider two numbers, represented by x and y, that satisfy the following conditions:
Seven times the first number, x, is added to the second number, y, to get 64.
The difference between three times the first number, x, and the second number, y, is 16.
Write an equation in terms of x and y that satisfies the first condition.
Write an equation in terms of x and y that satisfies the second condition.
Find the value of x and of y.
The length of a rectangle is 12 units more than the width, and the perimeter of the rectangle is 56 units.
Set up two equations by letting y be the width and x be the length of the rectangle.
Solve the system of equations.
When comparing some test results Christa noticed that the sum of her Geography score and Science score was 172, and that their difference was 18. She also noticed that she scored higher for the Geography test.
Let her Geography score be x and her Science score be y.
Write two equations in terms of x and y from the information given.
Use these two equations to find her Geography score.
Find her Science score.
A mother is currently 10 times older than her son. In 3 years time, she will be 7 times older than her son.
Set up two equations by letting x and y be the present ages of the son and mother respectively.
Solve the system of equations.
A man is five times as old as his son. Four years ago the man was nine times as old as his son. Let x and y be the ages of the man and his son respectively.
Write two equations relating x and y.
Find the ages of the man and his son.
Toby's piggy bank contains only 5\text{c} and 10\text{c} coins. It contains 70 coins with a total value of \$3.85.
Set up two equations by letting x and y be the number of 5\text{c} and 10\text{c} coins respectively.
Solve the system of equations.
Oprah invested \$16\,000 in total in two stocks A and B. In one year, the investment in stock A made a 14\% return, while the investment in stock B fell by 6\%. The total annual interest from both stocks was \$700.
Set up two equations be letting x and y be the amounts that she invested in stocks A and B respectively.
Solve the system of equations.
Explain the meaning of the solution of the system of equations.
A bank loaned out \$12\,000, part of it at a rate of 7\% per year and the rest at the rate of 13\% per year. The interest received for the year totalled \$1158.
Let x and y be the amounts, in dollars, that are loaned at the rates of 7\% and 13\% respectively.
Write two equations relating x and y.
Find the amounts the bank had loaned at a rate of 7\% and 13\%, to the nearest dollar.
Find the value of x and y in the following diagram:
Find the value of x and y in the following diagram:
Hence determine the length and width of the rectangle.
A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. After conducting some research, they discover that the cost of employing people to make the clothing is y = 800 + 60 x, where y is the cost and x is the number of t-shirts to be made, while the cost of using machinery (which includes the cost of purchasing the machines) is y = 3200 + 20 x.
Sketch a graph that depicts the two cost functions.
State the value of x, the number of t-shirts to be produced, at which it will cost the same whether the t-shirts are made by people or by machines.
State the range of values of x, the number of t-shirts to be produced, for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of x, the number of t-shirts to be produced, at which it will be more cost efficient to employ people to manufacture the t-shirts.
The plan for a proposed tunnel through a mountain side is shown in the following diagram:
The height of the mountain above sea level is given by the equation y = - \dfrac{x^{2}}{3} + 4, and the height of the tunnel above sea level is given by the equation y = \dfrac{x}{3} + 2, where y represents the height above sea level in kilometres.
Find the height above sea level at which cars will enter the tunnel, correct to two decimal places.