A hypothetical square grows at a rate of 16 m²/min. How fast are the sides of the square increasing when the sides are 15 m each?

Answer 1

The sides are increasing at a speed of #8/15# meters/minute.

The formula for area of a square is #A = s^2#, where #s# is the side length.
Differentiating #A# with respect to time:
#(dA)/dt = 2s((ds)/dt)#
Solve for #(ds)/dt#, since this represents the change in the sides with respect to time.
#((dA)/dt)/(2s) = (ds)/dt#
#1/(2s) xx (dA)/dt = (ds)/dt#

Here's what we know and what would be our unknown:

-We know the speed at which the area is changing (16 #m^2#/min) -We want to know the speed at which the lengths of our sides are changing at the moment when the sides are #15# meters each.
#1/(2 xx 15) xx 16 =(ds)/dt#
#1/30 xx 16 =( ds)/dt#
#8/15 = (ds)/dt#
Hence, the length of the sides are increasing at a speed of #8/15 "m"/"min"# when the sides are at length #15# meters.

Hopefully this helps!

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Answer 2

To find the rate at which the sides of the square are increasing, you can use the formula for the rate of change of area with respect to time. Since the area ( A ) of a square is given by ( A = s^2 ), where ( s ) is the length of a side, the rate of change of area ( \frac{dA}{dt} ) is ( 2s \frac{ds}{dt} ), where ( \frac{ds}{dt} ) is the rate at which the sides are increasing.

Given that ( \frac{dA}{dt} = 16 , \text{m}^2/\text{min} ) and ( s = 15 , \text{m} ), you can solve for ( \frac{ds}{dt} ):

[ \frac{dA}{dt} = 2s \frac{ds}{dt} ] [ 16 = 2(15) \frac{ds}{dt} ] [ 16 = 30 \frac{ds}{dt} ] [ \frac{ds}{dt} = \frac{16}{30} = \frac{8}{15} , \text{m/min} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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