Question

Give a reasonable domain for the function 10w-w^ where w is the width

^ means squared

Answer 1

Answer: 0 < w < 10

Step-by-step explanation

w^2 is the same as
`w^2`

10w-w^2 = 0

w(10-w) = 0

w = 0 or 10-w = 0

w = 0 or w = 10

Anything between w = 0 and w = 10, excluding each endpoint, will make 10w-w^2 positive. For example, if w = 5 then 10w-w^2 = 10*5-5^2 = 25.

Therefore, a reasonable domain would be
`0 < w < 10` which can be verbally described as "w is between 0 and 10 excluding both endpoints".

If we tried something like w = 12, then 10w-w^2 = 10*12-12^2 = -24, which wouldn't be a valid area value. Areas must be positive. I assume that 10-w is the length, so w(10-w) = width*length = area of rectangle.

We cannot have a width of w = 0 or w = 10 because it would lead to an area of 0.