Danica and Darrelle have developed a lucrative business selling handmade silver bracelets. Darrelle makes the more complex bracelets and Danica makes the simpler ones. It takes Darrelle 4 hours to make 3 bracelets and Danica can make 5 bracelets every 2 hours. a. Convert this scenario into linear equation (s); show both the standard form and the slope-intercept form of your equation (s). b. Could any part of this scenario be graphed using any of the special functions discussed in the course text? Explain your answer. c. Each bracelet Darrelle sells makes $10.50 in profit. Each bracelet Danica sells makes $3.50 in profit. If Darrelle and Danica work the same amount of time, explain each of the steps you will use and then calculate how many hours will it take for the whole business to make at least $500 profit?

a) a = (3/4) t

4a = 3t

b = (5/2) t

2b = 5t

b) See explanation below

c) It will take 30.075 hours for the whole business to make at least $500 profit

Step-by-step explanation:

a) First we would represent the information given in terms of amount of bracelet produced and time spent in production with variables in order to convert to linear equations

Let number of bracelet Darrelle makes = a

Number of bracelet Danica makes = b

Let the time spent in producing them in hours = t

Darrelle takes 4 hours to make 3 bracelets:

Rate of producing bracelet = (number of bracelet produced) / time

Rate = a/t = 3/4

4a = 3t

Linear equation form: y = mx + c

a = (3/4) t

The above equation is in the slope-intercept form indicating the intercept = 0 and slope = 3/4

4a - 3t = 0 (equation in standard form)

Danica makes 5 bracelets every 2 hours:

Rate of producing bracelet = (number of bracelet produced) / time

Rate = b/t = 5/2

2b = 5t

Linear equation form: y = mx + c

b = (5/2) t

The above equation is in the slope-intercept form indicating the intercept = 0 and slope = 5/2

2b - 5t = 0 (equation in standard form)

b) The equations are linear equations, hence they can be graphed.

But as regards graphing using special functions, this answer can only be answered by you as I'm not aware of the special functions discussed in the course.

c) Darrelle makes $10.50 in profit per bracelet.

Danica makes $3.50 in profit per bracelet

Since Darrelle and Danica work the same amount of time, we have to find the relationship between their profit, the number of bracelets produced and the time.

Profit for Darrelle for 'a' number of bracelet product = $10.50 * a = $10.50a

Profit for Danica per 'b' number of bracelet product = $3.50 * b = $3.50b

Let P = total profit made by both

P = $10.50a + $3.50b

Relationship of profit in terms of time spent in production when they work same amount of time:

P = $10.50 (3/4 * t) + $3.50 (5/2 * t)

P = 16.625t

When P = $500, t = ?

500 = 10.50 (3/4 * t) + 3.50 (5/2 * t)

500 = 7.875t + 8.75t

500 = 16.625t

t = 30.075 hours

It will take 30.075 hours for the whole business to make at least $500 profit