Question

banquet hall has two types of tables, rectangular and round. the rectangular tables can fit 8 people sitting around them. the round tables can fit 6 people sitting around them. the hall will be hosting a banquet for 192 people. assume all tables will have the maximum number of people seated at the table. a. write an equation in standard form showing the relationship between the number of rectangular tables, x, and the number of round tables, y, that are needed to seat 192 people. 8x + 6y = 192 b. find the x - intercept and explain its meaning in the context of the problem. c. find the y - intercept and explain its meaning in the context of the problem.

Explanation:

Step1: Write the standard - form equation

The number of people seated at rectangular tables is \(8x\) (since each rectangular table seats 8 people) and the number of people seated at round tables is \(6y\) (since each round table seats 6 people). The total number of people to be seated is 192. So the standard - form equation is \(8x + 6y=192\). We can simplify this equation by dividing through by 2 to get \(4x + 3y = 96\).

Step2: Find the x - intercept

The x - intercept is found by setting \(y = 0\) in the equation \(4x+3y = 96\).
When \(y = 0\), we have \(4x=96\), then \(x=\frac{96}{4}=24\). In the context of the problem, the x - intercept of 24 means that if no round tables are used (\(y = 0\)), 24 rectangular tables are needed to seat 192 people.

Step3: Find the y - intercept

The y - intercept is found by setting \(x = 0\) in the equation \(4x + 3y=96\).
When \(x = 0\), we have \(3y=96\), then \(y=\frac{96}{3}=32\). In the context of the problem, the y - intercept of 32 means that if no rectangular tables are used (\(x = 0\)), 32 round tables are needed to seat 192 people.

Answer:

a. The standard - form equation is \(4x + 3y=96\)
b. The x - intercept is 24. It means that if no round tables are used, 24 rectangular tables are needed to seat 192 people.
c. The y - intercept is 32. It means that if no rectangular tables are used, 32 round tables are needed to seat 192 people.