The paint mixer should mix 1.238 gallon paint that is 30 % gloss and 2.512 gallon paint that is 15 % gloss to get 3.75 gallon of paint that is 20 % gloss.
In this question we must determine the correct volumes of each paint to obtain a 20 % gloss with paint by means of weighted averages:
[tex]c_{r} = x\cdot c_{a} +(1-x)\cdot c_{b}[/tex] (1)
Where:
- [tex]x[/tex] - Proportion of the paint that is 30 % gloss.
- [tex]c_{a}[/tex] - Concentration ratio of the paint that is 30 % gloss.
- [tex]c_{b}[/tex] - Concentration ratio of the paint that is 15 % gloss.
- [tex]c_{r}[/tex] - Resulting concentration ratio
If we know that [tex]c_{a} = 0.3[/tex], [tex]c_{b} = 0.15[/tex] and [tex]c_{r} = 0.2[/tex], then the proportion of the paint that is 30 % gloss is:
[tex]0.2 = 0.3\cdot x +(1-x)\cdot 0.15[/tex]
[tex]0.2 = 0.15\cdot x+0.15[/tex]
[tex]0.15\cdot x = 0.05[/tex]
[tex]x = 0.33[/tex]
And the quantity of gallons needed of each paint are:
Paint (30 % gloss)
[tex]V_{a} = 0.33\cdot (3.75\,gal)[/tex]
[tex]V_{a} = 1.238\,gal[/tex]
Paint (15 % gloss)
[tex]V_{b} = V-V_{a}[/tex]
[tex]V_{b} = 3.75\,gal-1.238\,gal[/tex]
[tex]V_{b} = 2..512\,gal[/tex]
The paint mixer should mix 1.238 gallon paint that is 30 % gloss and 2.512 gallon paint that is 15 % gloss to get 3.75 gallon of paint that is 20 % gloss.
To learn more on weighted averages, we kindly invite to check this question: https://brainly.com/question/18554478
