Clare wants to make an open-top box by cutting out corners of a 30 inch by 25 inch piece of poster board and then folding up the sides. The volume, `V\left(x\right),` in cubic inches of the open-top box is a function of the side length, `x,`in inches, of the square cutouts.

V(x) = 4x³ - 110x² + 750x

Step-by-step explanation:

For each side of the base of the open-top box, since x inches is cut out from each side, we have 2x inches removed from each sides dimension.

Since the dimensions of the poster are 30 inches by 25 inches, the area of the base of the open-top box is thus

A(x) = (30 - 2x)(25 - 2x) since we remove 2x inches from each dimension.

Now, the height of the open-top box is x. So, its volume is V(x) = Ax

V(x) = (30 - 2x)(25 - 2x)x

V(x) = (750 - 60x - 50x + 4x²)x

V(x) = (750 - 110x + 4x²)x

Expanding the bracket, we have

V(x) = 750x - 110x² + 4x³

Re-arranging, we have

V(x) = 4x³ - 110x² + 750x

first option

a function is a relation in which an input is paired with exactly one output.

so, let's find the right answer.

in our first option, our inputs are january, february, and march. the outputs are 30, 28, and 25. each input has it's own output. this is correct!

our second option we know is incorrect, because ms. valverade has two outputs, dogs and cats. this is not a function.

our third option is incorrect as well. the yellow square has two ouputs, the square and the triangle.

lastly, in our fourth option, we see the number 7 two times with a different output. therefore, this is not a function.

remember: each input must be paired with only one output!