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3 years ago

13

Each square on Olivia's chessboard is 11 square centimeters.A chessboard has 8 squares on each sid.To the nearest tenth,what is

the width of Olivia's chessboard?

1 answer:

stellarik [79]3 years ago

5 0

First, find the length of one side of a square:
Area = side²
11 = s²
s = √(11) cm

Since there are 8 squares on each side, multiply the length of one square by 8 to get the length of 8 squares.
√(11) * 8 = 8√(11) ≈16.5 cm

Hope this helps!

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2 years ago

Let f(x) = 1/x^2 (a) Use the definition of the derivatve to find f'(x). (b) Find the equation of the tangent line at x=2

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Answer:

(a) $f'(x)=-\frac{2}{x^3}$

(b) $y=-0.25x+0.75$

Step-by-step explanation:

The given function is

$f(x)=\frac{1}{x^2}$ .... (1)

According to the first principle of the derivative,

$f'(x)=lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

$f'(x)=lim_{h\rightarrow 0}\frac{\frac{1}{(x+h)^2}-\frac{1}{x^2}}{h}$

$f'(x)=lim_{h\rightarrow 0}\frac{\frac{x^2-(x+h)^2}{x^2(x+h)^2}}{h}$

$f'(x)=lim_{h\rightarrow 0}\frac{x^2-x^2-2xh-h^2}{hx^2(x+h)^2}$

$f'(x)=lim_{h\rightarrow 0}\frac{-2xh-h^2}{hx^2(x+h)^2}$

$f'(x)=lim_{h\rightarrow 0}\frac{-h(2x+h)}{hx^2(x+h)^2}$

Cancel out common factors.

$f'(x)=lim_{h\rightarrow 0}\frac{-(2x+h)}{x^2(x+h)^2}$

By applying limit, we get

$f'(x)=\frac{-(2x+0)}{x^2(x+0)^2}$

$f'(x)=\frac{-2x)}{x^4}$

$f'(x)=\frac{-2)}{x^3}$ .... (2)

Therefore $f'(x)=-\frac{2}{x^3}$ .

(b)

Put x=2, to find the y-coordinate of point of tangency.

$f(x)=\frac{1}{2^2}=\frac{1}{4}=0.25$

The coordinates of point of tangency are (2,0.25).

The slope of tangent at x=2 is

$m=(\frac{dy}{dx})_{x=2}=f'(x)_{x=2}$

Substitute x=2 in equation 2.

$f'(2)=\frac{-2}{(2)^3}=\frac{-2}{8}=\frac{-1}{4}=-0.25$

The slope of the tangent line at x=2 is -0.25.

The slope of tangent is -0.25 and the tangent passes through the point (2,0.25).

Using point slope form the equation of tangent is

$y-y_1=m(x-x_1)$

$y-0.25=-0.25(x-2)$

$y-0.25=-0.25x+0.5$

$y=-0.25x+0.5+0.25$

$y=-0.25x+0.75$

Therefore the equation of the tangent line at x=2 is y=-0.25x+0.75.

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3 years ago

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1 year ago

Your brother is 1313 your age. Your sister is 6 years older than your brother. Your sister is 10 years old. Write and solve an e

il63 [147K]

There seems to be a typo. You wrote 1313, but I'm assuming you meant to say the fraction 1/3. If this assumption is correct, then continue with the rest of the solution. If the assumption is not correct, then ignore what I wrote below and please provide clarification of what the "1313" should represent. Thank you.

----------------

B = your brother's age

S = your sister's age

Y = your age

All of these ages are current present day ages.

---------------

"your brother is 1/3 your age" so B = (1/3)Y or B = Y/3.

"Your sister is 6 years older than your brother" so S = B+6

"Your sister is 10 years old" so S = 10.

---------------

S = B+6

B+6 = S

B+6 = 10 ... plug in S = 10

B+6-6 = 10-6 ... subtract 6 from both sides

B = 4

Your brother is 4 years old.

---------------

B = Y/3

Y/3 = B

Y/3 = 4 .... plug in B = 4

3(Y/3) = 3*4 ... multiply both sides by 3

Y = 12

<h3>Answer: Your age is 12 years old.</h3>

note: This answer hinges on assumption that the "1313" should be "1/3".

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3 years ago