Question

PLEASE HELP PLEASE!!

Kim owns a chocolate company. She developed a new method of making a certain type of chocolate that she thinks will be faster than her original method. Original method can be represented by this function: g(x) = 65(1.1)^x. Her new method is shown in the graph below.

a. During week 0, how many chocolates were produced using the old method? Show your work.
b. During week 0, how many chocolates were produced using the new method? Explain how you know.
c. What is the positive difference between the old method and new method? Show your work.
d. What method initially made more chocolate? Explain how you know or show your work.
e. How many chocolates did the new method produce during week 3.
f. How many chocolates did the old method produce during week 3.
g. In week 3, which method produced more chocolates?

Answer 1

The new method and the old method are illustrations of exponential functions.

• 65 chocolates were produced using the old method in week 0
• 60.281 chocolates were produced using the new method in week 0
• The positive difference between both methods is 0.1
• The old method made more initially.
• 104.167 chocolates were produced using the new method in week 3
• 86.515 chocolates were produced using the old method in week 3
• The new method made more in week 3.

(a) Old method week 0

The old method is given as:

$\mathbf{g(x) = 65(1.1)^x}$

At week 0, x = 0

So, we have:

$\mathbf{g(0) = 65(1.1)^0= 65}$

65 chocolates were produced using the old method in week 0

(b) New method week 0

At week 0, x = 0

From the graph

$\mathbf{y = 60.282\ when\ x = 0}$

60.281 chocolates were produced using the new method in week 0

(c) The positive difference between the methods

An exponential function is represented as:

$\mathbf{y = ab^x}$

Where: b represents rate

For the old method,

$\mathbf{b =1.1}$

For the new method, we have:

$\mathbf{(x_1,y_1) = (0,60.282)}$

$\mathbf{(x_2,y_2) = (1,72.337)}$

So, we have:

$\mathbf{y = ab^x}$

$\mathbf{ab^0 = 60.282}$

$\mathbf{a = 60.282}$

$\mathbf{y = ab^x}$

$\mathbf{ab^1 = 72.337}$

Substitute $\mathbf{a = 60.282}$

$\mathbf{60.282 \times b=72.337}$

Divide both sides by 60.282

$\mathbf{ b=1.20}$

So, the positive difference between both methods is their rate.

The difference in their rates is:

$\mathbf{d = |b_2 - b_1|}$

$\mathbf{d = |1.20 - 1.1|}$

$\mathbf{d = 0.1}$

Hence, the positive difference between both methods is 0.1

(d) Which made more initially

In (a), we have:

$\mathbf{g(0) = 65}$

In (b), we have:

$\mathbf{y = 60.282}$

These values represent the initial chocolate

Hence, the old method made more initially.

(e) New method week 3

At week 3, x = 3

From the graph

$\mathbf{y = 104.167\ when\ x = 3}$

104.167 chocolates were produced using the new method in week 3

(f) Old method week 3

The old method is given as:

$\mathbf{g(x) = 65(1.1)^x}$

At week 3, x = 3

So, we have:

$\mathbf{g(3) = 65(1.1)^3= 86.515}$

86.515 chocolates were produced using the old method in week 3

(f) Which made more initially

In (e), we have:

$\mathbf{y = 104.167\ when\ x = 3}$

In (f), we have:

$\mathbf{g(3) = 86.515}$

These values represent the amount of chocolate made in the week 3

Hence, the new method made more in week 3.

Read more about exponential functions at:

https://brainly.com/question/3127939