Question

A school will spend at most $5,200 to buy new computer equipment for their computer lab. each new monitor will cost $190 and each new keyboard will cost $75.

Answer 1

To find the maximum number of monitors and keyboards the school can buy with a budget of $5,200, set up an inequality and solve algebraically.

Step-by-step explanation:

To find the maximum number of monitors and keyboards the school can buy with a budget of $5,200, we need to set up an inequality. Let x represent the number of monitors and y represent the number of keyboards.

Given the cost of each monitor is $190 and each keyboard is $75, the inequality can be written as:

190x + 75y ≤ 5200

We can solve this inequality by graphing or using a table of values. However, since this is a mathematics question, it is more appropriate to solve it algebraically.

By rearranging the inequality and solving for y, we can express y in terms of x:

y ≤ (5200 - 190x) / 75

This equation shows that for every monitor the school purchases, the number of keyboards must be less than or equal to (5200 - 190x) / 75.

Answer 2

The school can buy a maximum of 27 monitors and 32 keyboards with the $5,200 budget for new computer equipment. Therefore, the total cost of the monitors will be $5,110 ($190 per monitor x 27 monitors) and the total cost of the keyboards will be $2,400 ($75 per keyboard x 32 keyboards). The remaining $690 can be used for other necessary expenses such as cables, mouse, and software.

Step-by-step explanation:

To maximize the use of the $5,200 budget for new computer equipment, we need to find the number of monitors and keyboards that can be bought. Let's assume that x monitors and y keyboards are bought. The total cost of the monitors will be x times $190, and the total cost of the keyboards will be y times $75. The remaining amount will be used for other expenses.

Using mathematical notation, we can represent this as follows:

x * 190 + y * 75 + Other expenses = 5200

We can simplify this equation by finding the value of x and y that satisfy this equation. Since we want to find the maximum number of monitors and keyboards that can be bought, we will find the values of x and y that result in the highest possible number of monitors and keyboards.

To do this, we can use calculus to find the critical points of this function. The derivative of this equation with respect to x is:

190 - 190 * (x/5200)^(-1) * (y/5200)^(-1) * (5200 - x - y) / (5200 - x)

The derivative with respect to y is:

75 - 75 * (x/5200)^(-1) * (y/5200)^(-1) * (5200 - x - y) / (5200 - y)

To find the critical points, we need to set these derivatives equal to zero and solve for x and y. After simplifying these equations, we get:

x = 487.5 and y = 387.5

However, since x and y cannot be negative or decimal numbers in this context, we round them up to get:

x = 488 and y = 388

Therefore, we can buy a maximum of 488 monitors and 388 keyboards with the $5,200 budget for new computer equipment. This results in a total cost of $96,640 for the monitors ($190 per monitor x 488 monitors) and $29,440 for the keyboards ($75 per keyboard x 388 keyboards). The remaining $690 can be used for other necessary expenses such as cables, mouse, and software.