<img src="https://tex.z-dn.net/?f=n%5E%7B2%7D%20%20%7Bn%7D%5E%7B2%7D%20" id="TexFormula1" title="n^{2} {n}^{2} " alt="n^{2} {n

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kirill115 [55]

3 years ago

}^{2} " align="absmiddle" class="latex-formula">

Mathematics

1 answer:

nadya68 [22]3 years ago

8 0

1 Use Product Rule: {x}^{a}{x}^{b}={x}^{a+b}xaxb=xa+b
{n}^{2+2}n2+2
2 Simplify 2+22+2 to 44
{n}^{4}
answer is n4

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The distance between Lincoln, NE, and Boulder, CO, is about 500 miles. The distance between Boulder, CO, and a third city is 200

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We know that

The sum of the lengths of any two sides of a triangle is greater than the length of the third side (Triangle Inequality Theorem)

Let

A------> Lincoln, NE

B------> Boulder, CO

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we know that

in the triangle ABC

AB=500 miles

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Applying the Triangle Inequality Theorem

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A homeowner plans to enclose a 200 square foot rectangular playground in his garden, with one side along the boundary of his pro

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Answer:

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 $=\$(\frac13x)$

Now homeowner's total cost for the material is

$=\$[ 2(x+y)-\frac13x]$

$=\$[2x+2y-\frac13x]$

$=\$[2y+x+x-\frac13x]$

$=\$[2y+x+\frac{3x-x}{3}]$

$=\$[2y+x+\frac{2}{3}x]$

$=\$[2y+\frac53x]$

$\therefore C(x)=2y+\frac53x$

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

$\Rightarrow y=\frac{200}{x}$

Putting the value of y in C(x)

$\therefore C(x)=2(\frac{200}{x})+\frac53x$

The domain of C is $(0,\infty )$ .

$\therefore C(x)=2(\frac{200}{x})+\frac53x$

Differentiating with respect to x

$C'(x)= - \frac{400}{x^2}+\frac53$

Again differentiating with respect to x

$C''(x) = \frac{800}{x^3}$

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

$\therefore 0= - \frac{400}{x^2}+\frac53$

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Therefore

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

Putting the value of x in $y=\frac{200}{x}$ we get

$\therefore y=\frac{200}{15.5}$

=12.9

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

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