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3 years ago

What is 3 1/4 - (-5/8)

Mathematics

1 answer:

Aleks04 [339]3 years ago

6 0

Answer:

First we need to find a common denominator;

3 1/4 and -5/8 have a common denominator of 8

-5/8 already has 8 now we alter 3 1/4 to have 8 as the denominator as well.

if we make 3 1/4 into an improper fraction we would get 13/4

13/4 x 2/2 = 26/8, because 2 x 4 = 8 so we also apply that to the numerator

Now we have;

26/8 - (-5/8)

the two negatives combine to make a positive:

26/8 + 5/8

= 31/8 which simplifies to

3 7/8

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A circle is growing so that the radius is increasing at the rate of 3 cm/min. How fast is the area of the circle changing at the

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The area is growing at a rate of $\frac{dA}{dt} =226.2 \,\frac{cm^2}{min}$

Step-by-step explanation:

<em>Notice that this problem requires the use of implicit differentiation in related rates (some some calculus concepts to be understood), and not all middle school students cover such.</em>

We identify that the info given on the increasing rate of the circle's radius is 3 $\frac{cm}{min}$ and we identify such as the following differential rate:

$\frac{dr}{dt} = 3\,\frac{cm}{min}$

Our unknown is the rate at which the area (A) of the circle is growing under these circumstances,that is, we need to find $\frac{dA}{dt}$ .

So we look into a formula for the area (A) of a circle in terms of its radius (r), so as to have a way of connecting both quantities (A and r):

$A=\pi\,r^2$

We now apply the derivative operator with respect to time ( $\frac{d}{dt}$ ) to this equation, and use chain rule as we find the quadratic form of the radius:

$\frac{d}{dt} [A=\pi\,r^2]\\\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}$

Now we replace the known values of the rate at which the radius is growing ( $\frac{dr}{dt} = 3\,\frac{cm}{min}$ ), and also the value of the radius (r = 12 cm) at which we need to find he specific rate of change for the area :

$\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}\\\frac{dA}{dt} =\pi\,*2*(12\,cm)*(3\,\frac{cm}{min}) \\\frac{dA}{dt} =226.19467 \,\frac{cm^2}{min}\\$

which we can round to one decimal place as:

$\frac{dA}{dt} =226.2 \,\frac{cm^2}{min}$

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