Optimal coverage question need help.

[Mathfan101]

You are configuring a machine that mixes water-based paint. The proportion of water in the paint determines its viscosity (thickness), and therefore is a major factor in the expected coverage per tin of paint.

With no water, the paint is too thick to spread evenly, so the expected coverage is low. Adding water reduces the viscosity of the paint, enabling it to spread more evenly and achieve greater coverage. However, if too much water is added, the paint becomes too thin and coverage is reduced.

You have determined that the expected coverage for a tin of paint can be calculated by the following function, in which x represents the number of centileters (cl) of water added to a tin:
c(x)=−2x^2+30x+3

How many centiliters of water you should configure the machine to add to the paint in order to achieve optimal coverage?

My thoughts below:

I've calculated the local maxima is (7.5, 115.5)
So 115.5 is c(x) which is the water required to reach optimum coverage. Can you please advise where I was wrong?

 

[tkhunny]

Read your definition more carefully: "x represents the number of centileters (cl) of water added to a tin"

"maxima" is plural
"maximum" is singular

 

[Mathfan101]

Thanks @tkhunny
Does this mean 7.5 is the right answer for this question? Because it represent the local maximum for x?

 

[Deleted member 4993]

You are configuring a machine that mixes water-based paint. The proportion of water in the paint determines its viscosity (thickness), and therefore is a major factor in the expected coverage per tin of paint.

With no water, the paint is too thick to spread evenly, so the expected coverage is low. Adding water reduces the viscosity of the paint, enabling it to spread more evenly and achieve greater coverage. However, if too much water is added, the paint becomes too thin and coverage is reduced.

You have determined that the expected coverage for a tin of paint can be calculated by the following function, in which x represents the number of centileters (cl) of water added to a tin:
c(x)=−2x^2+30x+3

How many centiliters of water you should configure the machine to add to the paint in order to achieve optimal coverage?

My thoughts below:

I've calculated the local maxima is (7.5, 115.5)
So 115.5 is c(x) which is the water required to reach optimum coverage. Can you please advise where I was wrong?

How do you know that your answer is wrong?

 

[tkhunny]

Thanks @tkhunny
Does this mean 7.5 is the right answer for this question? Because it represent the local maximum for x?

?? x = 7.5 is not a "local maximum for x". That doesn't make any sense. Be more careful with your language.

x = 7.5 is the x-value (water) that gives the maximum for c(x) (the coverage).

It is "local", but in this case it is also global. That's how 2nd degree functions with a negative leading coefficient work.

So also it is incorrect to say "115.5 is c(x) which is the water required to reach optimum coverage".
c(x) is NOT the water. It is the coverage. Watch your words very carefully. Read your definitions several time in order to make them sink into your brain. That's one good reason to WRITE DOWN your definitions. It gives a hard reference so your words don't wander from what you mean.