Area: Change in Dimensions
A rectangle FGHJ has a width of 3 inches and a length of 7 inches.

Step-by-step explanation:

this is not complicated at all. first and second graders can do a-c (come on, you know how to calculate the area of a rectangle) and with some imagination also d and e.

this is meant for you to actually do it so that you get a feeling for these things.

I will therefore leave the pure calculation to you.

and I will give you some help for the conceptual/ theoretical part.

the area of a rectangle is, of course, length × width (or height some on the view you have on an object).

let's just call them l and w.

so the regular area is

Ar = l×w

now one dimension (or example and for the width, but it does not matter, it is the same for the length) is doubled.

it simply means the new width is twice the old width.

so the area with a double width is

A2w = l×(2×w) = 2×(l×w)

so, that means also the area is doubled compared to the regular area.

now, both dimensions are doubled.

A2w2l = (2×l)×(2×w) = 4×(l×w)

so, the area grew 4 times a large as the regular area.

what is the rule here ?

let's triple the dimensions.

A3w3l = (3×l)×(3×w) = 9×(l×w)

as you can easily see : the new area grows with the square of the scaling factor, when we change both dimensions equally.

the reason is simple - as our little equations show, we have to multiply in the scaling factor of every dimension in play.

therefore, the new area grows with the product of the scaling factors of the involved dimensions. if both scaling factors are the same, then the area grows therefore by the square of the common factor.

and this principle applies to every shape, where the area is constructed out of 2 dimensions. any other additional multiplication factors don't matter for this, because this is all one big multiplication of several factors, and if you multiply one of these factors by an additional factor, the whole result changes by that factor.

step-by-step explanation:

we have

using a graphing tool

see the attached figure

this is a quadratic equation (vertical parabola) open up

the vertex is a > vertex

the domain of the function is all real > interval (-∞,∞)

the range of the function is the > [-6.125,∞)

the function has two x->   and

the function is increasing over the > (0.25,∞)

the function is decreasing over the > (-∞,0.25)