The relationship between the percentage of frozen citrus crop, and the cost of box of oranges is an illustration of a linear function.
- <em>The linear equation of the function is: </em>
<em>.</em> - <em>The inverse function is: </em>
<em>
.</em> - <em>A practical domain is from 0% to 100%</em>
- <em>A practical range is from 7 to 29.9
</em>
<u>A. Input quantity</u>
The input quantity is the percentage of frozen citrus crop
<u />
<u>B. Output quantity
</u>
The output quantity is the cost of box of oranges
<u>C. The linear function</u>
We have:
![(P_1,c_1) = (20\%,11.58)\\(P_2,c_2) = (80\%,25.32)](https://tex.z-dn.net/?f=%28P_1%2Cc_1%29%20%3D%20%2820%5C%25%2C11.58%29%5C%5C%28P_2%2Cc_2%29%20%3D%20%2880%5C%25%2C25.32%29)
<em>Calculate the slope of the function</em>
![m = \frac{c_2 - c_1}{P_2 - P_1}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7Bc_2%20-%20c_1%7D%7BP_2%20-%20P_1%7D)
![m = \frac{25.32 - 11.58}{80\%-20\%}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B25.32%20-%2011.58%7D%7B80%5C%25-20%5C%25%7D)
![m = \frac{13.74}{60\%}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B13.74%7D%7B60%5C%25%7D)
![m = 22.9](https://tex.z-dn.net/?f=m%20%3D%2022.9)
<em>The linear equation is calculated as follows:</em>
![c -c_1 = m(P-P_1)](https://tex.z-dn.net/?f=c%20-c_1%20%3D%20m%28P-P_1%29)
![c -11.58= 22.9(P-20\%)](https://tex.z-dn.net/?f=c%20-11.58%3D%2022.9%28P-20%5C%25%29)
![c-11.58 = 22.9P-4.58](https://tex.z-dn.net/?f=c-11.58%20%3D%2022.9P-4.58)
<u>D. Rewrite as y = mx + b</u>
We have:
![c-11.58 = 22.9P-4.58](https://tex.z-dn.net/?f=c-11.58%20%3D%2022.9P-4.58)
Collect like terms
![c = 22.9P - 4.58 + 11.58](https://tex.z-dn.net/?f=c%20%3D%2022.9P%20-%204.58%20%2B%2011.58)
![c = 22.9P+7](https://tex.z-dn.net/?f=c%20%3D%2022.9P%2B7)
<em>The function is:</em>
![g(P) = 22.9P+7](https://tex.z-dn.net/?f=g%28P%29%20%3D%2022.9P%2B7)
<u>E. A practical domain</u>
The domain is the possible values of P. Because P is a percentage, its possible values are 0% to 100%.
The domain of the function is: ![[0\%,100\%]](https://tex.z-dn.net/?f=%5B0%5C%25%2C100%5C%25%5D)
<u>F. A practical range</u>
When P = 0%
![c = 22.9 \times 0\% + 7 = 7](https://tex.z-dn.net/?f=c%20%3D%2022.9%20%5Ctimes%200%5C%25%20%2B%207%20%3D%207)
When P = 100%
Hence, the range of the function is: ![[7,29.9]](https://tex.z-dn.net/?f=%5B7%2C29.9%5D)
G. The meaning of ![g^{-1}(12)](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29)
The inverse function of g(P) is ![g^{-1}(P)](https://tex.z-dn.net/?f=g%5E%7B-1%7D%28P%29)
So:
is the percentage of frozen citrus crop, when the cost is $12.
<u>H. The inverse formula</u>
We have:
![c = 22.9P+7](https://tex.z-dn.net/?f=c%20%3D%2022.9P%2B7)
Subtract 7 from both sides
![c - 7 = 22.9P](https://tex.z-dn.net/?f=c%20-%207%20%3D%2022.9P)
Make P the subject
![P = \frac{1}{22.9}(c - 7)](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B1%7D%7B22.9%7D%28c%20-%207%29)
So, the inverse formula is:
![g^{-1}(c) = \frac{1}{22.9}(c - 7)](https://tex.z-dn.net/?f=g%5E%7B-1%7D%28c%29%20%3D%20%5Cfrac%7B1%7D%7B22.9%7D%28c%20-%207%29)
Substitute 12 for c
![g^{-1}(12) = \frac{1}{22.9}(12 - 7)](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%20%5Cfrac%7B1%7D%7B22.9%7D%2812%20-%207%29)
![g^{-1}(12) = \frac{1}{22.9} \times 5](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%20%5Cfrac%7B1%7D%7B22.9%7D%20%5Ctimes%205)
![g^{-1}(12) = 22\%](https://tex.z-dn.net/?f=g%5E%7B-1%7D%2812%29%20%3D%2022%5C%25)
Read more about linear equations at:
brainly.com/question/19770987