Question

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?

Answer 1

To address the student's question, linear equations for Danica's and Darrelle's bracelet-making rates were derived in both standard form and slope-intercept form. The scenario does not involve any special functions such as exponential or quadratic relationships. Finally, by calculating the combined profit per hour made by both Danica and Darrelle, it was determined that approximately 30.08 hours of work are needed to achieve at least a $500 profit.

Step-by-step explanation:

To convert the scenario into linear equations, let us consider the number of bracelets as the dependent variable (y) and the time in hours as the independent variable (x). For Darrelle, the rate at which bracelets are made is 3 bracelets per 4 hours, so the equation is:

Darrelle's rate: y = (3/4)x

This can also be expressed in slope-intercept form as:

y = 0.75x

For Danica, the rate at which bracelets are made is 5 bracelets per 2 hours, so the equation is:

Danica's rate: y = (5/2)x

This can also be written in slope-intercept form as:

y = 2.5x

Now, considering the special functions, this scenario cannot be graphed using such as the relationship between time and the number of bracelets made is linear, not a special function like exponential or quadratic.

To calculate the number of hours needed to make at least $500 in profit, you first determine the hourly profit by multiplying the number of bracelets made per hour by their respective profit:

Darrelle's hourly profit = (3/4) * $10.50 = $7.875

Danica's hourly profit = (5/2) * $3.50 = $8.75

The total hourly profit when both Danica and Darrelle work together is:

Total hourly profit = Darrelle's hourly profit + Danica's hourly profit

Total hourly profit = $7.875 + $8.75 = $16.625

To find the hours needed for a $500 profit, divide the target profit by the total hourly profit:

Hours needed = $500 / $16.625 ≈ 30.08

So, they would need to work for approximately 30.08 hours to make at least $500 in profit.

Answer 2

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