I am a painter. I want the price of my paintings to scale in a linear fashion, but don't want to charge a set amount per square cm. I want the price to decrease as the painting size increases in a straight, linear line.

My cheapest painting is £395 for a 28 x 23cm painting with an area of 644cm2

My most expensive painting is £2950 for a 90 x 90cm painting with an area of 8100cm2

Please tell me the formula I can use in excel to input any value for the cm squared area, that tells me the price for that size.

Clearly this is not a problem for a class, so it's ok for me to just tell you the answer, if I can.

We'll have to clarify what you want, but we can figure it out one way or another! And yes, it is basic algebra.

You say you want the

*price *to decrease as size increases; clearly the actual

*cost *of a painting

*increases*, so you must be referring to the

*price per square centimeter* decreasing linearly. Is that what you have in mind?

Alternatively, when you say you want the price to scale linearly, you could mean that you want the cost of a painting to be a linear function of the area, which would be a different thing. I'll assume that is not what you want. It would amount to charging a fixed price per cm

^{2} plus some overhead.

First, the price per square cm of the 644 cm

^{2} painting is 0.613, while for the 8100 cm

^{2} it is 0.364. Using the point-slope formula, this gives the price per cm

^{2} as

P = - 0.0000334(A - 644) + 0.613

so

P = 0.6345 - 0.0000334A

Then the cost is the price times the area, giving

C = (0.6345 - 0.0000334A)A = 0.6345A - 0.0000334A

^{2}
As you see, when the price per cm

^{2} increases linearly, the cost increases quadratically. This implies it will rise to a peak cost, after which the cost of larger paintings would actually decrease. In fact, the price per cm

^{2}, by my formula for P, actually reaches zero at about 19,000 cm

^{2}, and then goes negative. Presumably you'll never make one that large? That's the danger of decreasing prices linearly.

Does this look like what you want?