Question

1. maura has 2 hours on a saturday to do chores. it takes her 15 minutes to fold a load of laundry. it takes her 20 minutes to dust a room and it takes her 15 minutes to fold a load of laundry. a. write an equation in standard form for the line that gives all the possible number of rooms she can dust, x, and the loads of laundry she can fold, y, in a 2 - hour time span. b. find the x - intercept of this line and interpret the value in the context of this problem. c. interpret the slope of your equation in the context of this problem.

Explanation:

Step1: Convert time to minutes

2 hours = 120 minutes.

Step2: Set up the equation

Let \(x\) be the number of rooms dusted and \(y\) be the number of loads of laundry folded. The time - constraint equation based on the time it takes for each task is \(20x + 15y=120\). We can simplify this equation by dividing through by 5, getting \(4x + 3y = 24\). This is the standard - form equation (\(Ax+By = C\), where \(A = 4\), \(B = 3\), and \(C = 24\)).

Step3: Find the x - intercept

To find the \(x\) - intercept, set \(y = 0\) in the equation \(4x+3y = 24\). Then \(4x=24\), so \(x = 6\). In the context of the problem, the \(x\) - intercept of 6 means that if Maura does not fold any laundry (\(y = 0\)), she can dust 6 rooms in 2 hours.

Step4: Interpret the slope

Rewrite the equation \(4x + 3y=24\) in slope - intercept form \(y=mx + b\). First, solve for \(y\): \(3y=-4x + 24\), so \(y=-\frac{4}{3}x + 8\). The slope \(m=-\frac{4}{3}\). In the context of the problem, the slope means that for every additional room Maura dusts (\(x\) increases by 1), the number of loads of laundry she can fold (\(y\)) decreases by \(\frac{4}{3}\). Since we are dealing with whole numbers of tasks in a real - world context, it implies a trade - off between dusting rooms and folding laundry within the 2 - hour time span.

Answer:

a. The standard - form equation is \(4x + 3y=24\).
b. The \(x\) - intercept is 6. It means that if Maura does not fold any laundry, she can dust 6 rooms in 2 hours.
c. The slope of the line \(y =-\frac{4}{3}x + 8\) is \(-\frac{4}{3}\). It means that for every additional room Maura dusts, the number of loads of laundry she can fold decreases by \(\frac{4}{3}\), representing a trade - off between the two tasks within the 2 - hour time limit.