t.
A rectangular playground space is to be fenced in using the wall of a daycare building
for one side and 200 meters of fencing for the other three sides. The area A(x) in
square meters of the playground space is a function of the length x in meters of each of
the sides perpendicular to the wall of the daycare building.
a.
What is the area of the playground when x = 50?
b.
Write an expression for A(x).
c.
What is a reasonable domain for A in this context?

b) The remaining fence, after the two sides of length x are fenced, is 200-2x. That is the length of the side parallel to the building. The product of the lengths parallel and perpendicular to the building is the area of the playground:

A(x) = x(200 -2x)

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a) A(50) = 50(200 -2·50) = 50·100 = 5000 . . . . m²

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c) The equation makes no sense if either length (x or 200-2x) is negative, so a reasonable domain is (0, 100). For x=0 or x=100, the playground area is zero, so we're not concerned with those cases, either. Those endpoints could be included in the domain if you like.

facundo3141592 facundo3141592 · Answer 2 · 2021-09-22T16:25:47+02:00

We know that for a rectangle of length L and width W the area is just given by:

A = L*W.

And the perimeter is:

P = 2*(L + W)

The solutions are:

a) 500 square meters.

b) A(x) = x*(200m - 2*x)

c) 0m < x < 100m

Here we have a rectangle of a perimeter equal to 200m + L

Where L is the side that is along the wall, and we call "x" as the length of the side perpendicular to the wall.

Then we also have:

200m = 2*x + L

a) If x = 50, we can use the above equation to find the length of the other side.

200m = 2*50m + L

200m = 100m + L

200m - 100m = L

100m = L

Then the two measures of the rectangle are:

L = 100m

x = 50m

Then the area is:

A = 100m*50m = 500m^2

b) Now we want a general expression for A(x).

We have:

A(x) = x*L

And we know that:

200m = 2*x + L

Then we can isolate L to get:

200m - 2*x =L

Now we can replace that in the area equation:

A(x) = x*(200m - 2*x)

Now the equation only depends on the variable x, as wanted.

c) The domain of a function is the set of possible inputs that we can use.

Here, x represents the length of one of the sides of a rectangle, so we should start with:

x > 0m

So x is a positive number.

Now let's go to the equation:

200m - 2*x =L

L is also a measure, so L also must be larger than zero meters, then we can write:

200m - 2*x > 0m

Now we can solve this for x:

200m > 2*x

200m/2 > x

100m > x

Writing together the two inequalities, we get:

0m < x < 100m

This is the domain of the area function.

If you want to learn more, you can read:

https://brainly.com/question/21982865