Which of the following sets of numbers could represent the three sides of a right
triangle?
{44, 60, 75}
{30, 73, 78}
{37, 77,85}
{39, 52, 65}

"{39, 52, 65}"

Step-by-step explanation:

Whenever there is a right-triangle, we can use the Pythagorean Theorem to find this answer

Pythagorean Theorem tell us that:

Leg^2 + Another Leg^2 = Hypotenuse^2

Note: Hypotenuse is the longest side

So, for each set, we can check if the Pythagorean Theorem holds.

{44, 60, 75}

75^2 = 5625

and

44^2 + 60^2 = 5536

This is not correct.

{30, 73, 78}

78^2 = 6084

30^2 + 73^2 = 6229

This is not correct as well.

{37, 77,85}

85^2 = 7225

37^2 + 77^2 = 7298

This isn't correct.

{39, 52, 65}

65^2 = 4225

and

39^2 + 52^2 = 4225

Here, pythagorean theorem holds, so this is the correct set.

Therefore the length and width of the playground are 15.5 feet and 12.9 feet respectively.

Step-by-step explanation:

Given that, a homeowner plans to enclose a 200 square foot rectangle playground.

Let the width of the playground be y and the length of the  playground be x which is the side along the boundary.

The perimeter of the playground is = 2(length +width)

                                                          =2(x+y) foot

The material costs $1 per foot.

Therefore total cost to give boundary of the play ground

=$[ 2(x+y)×1]    

=$[2(x+y)]  

But the neighbor will play one third of the side x foot.

So the neighbor will play  

Now homeowner's total cost for the material is

where C(x) is total cost of material in $.

Given that the area of the playground is 200.

We know that,

The area of a rectangle is =length×width

                                           =xy square foot

∴xy=200

Putting the value of y in C(x)

The domain of C is.

Differentiating with respect to x

Again differentiating with respect to x

To find the critical point set C'(x)=0

Therefore

Therefore at x= 15.5 , C(x) is minimum.

Putting the value of x in we get

    =12.9

Therefore the length and width of the playground are 15.5 feet and 12.9 feet respectively.