Why does multiplying the input of a linear function change
1 answer:
Answer:
Step-by-step explanation:
If we have a linear function and we multiply the input of the function, we aren't really <em>changing</em> the values of the function, we're changing the rate at which those values are reached.
E.g. if you look at the attached picture, we have and
, which is just
. The value when the red line is x = 1 is half that of the value of the blue line when x = 1.
The values of the f(2x) line are reaching the values of f(x) twice as fast, meaning it has a slope of twice as much. However, it still grows through the same intercept.
If you're interested in the calculus behind this, look up the chain rule of derivatives.
However, when you start changing the output of the function, you're changing the entire equation. It's not the same line anymore.
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Answer:
The dimensions that minimize the surface are:
Wide: 1.65 yd
Long: 3.30 yd
Height: 2.20 yd
Step-by-step explanation:
We have a rectangular base, that its twice as long as it is wide.
It must hold 12 yd^3 of debris.
We have to minimize the surface area, subjet to the restriction of volume (12 yd^3).
The surface is equal to:
The volume restriction is:
If we replace h in the surface equation, we have:
To optimize, we derive and equal to zero:
Then, the height h is:
The dimensions that minimize the surface are:
Wide: 1.65 yd
Long: 3.30 yd
Height: 2.20 yd
