Mathematics, 29.10.2019 06:31 lilianna25

The area and circumference of a circle of radius r are given by a(r) = πr2 and c(r) = 2πr. suppose we increase
the radius of a sphere by 0.001 units from r to r + 0.001.
a. use the binomial theorem to write an expression for the increase in area volume a(r + 0.001) − a(r) as a
sum of three terms.
b. write an expression for the average rate of change of the area as the radius increases from r to r + 0.001.
c. simplify the expression in part (b) to compute the average rate of change of the area of a circle as the radius
increases from r to r + 0.001.
d. what does the expression from part (c) resemble?
e. why does it make sense that the average rate of change should approximate the area of a circle? think about
the geometric figure formed by a(r + 0.001) − a(r). what does this represent?
f. how could we approximate the area of the shell using circumference? and the average rate of change for the
area?
g. find the difference between the average rate of change of the area and c(r) when r = 1.

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Answer from: seannalove4148

Please see the result step by step, as follows:

Step-by-step explanation:

Let us start by writing down both expressions for an given increased radius

$r'=r+\Delta r$ , where the second term is the radius increment:

Increased area: $A'=\pi (r+\Delta r)^{2} =\pi(r^2+(\Delta r)^2+2r\Delta r)=\pi r^2+ \pi(\Delta r)^2+2\pi r\Delta r$ Increased circumference (length): $C'=2 \pi (r + \Delta r) = 2 \pi r + 2 \pi \Delta r$

Here, he just used the binomial theorem, set as:

(a+b)^2=a^2+2ab+b^2 , applied to the sum of the new, increased ratio.

a. So, for an increment of $\Delta r = 0.001$ ,

$\Delta A(\Delta r = 0.001) = A(r+0.001) - A(r) = (\pi r^2+ \pi(\Delta r)^2+2\pi r \Delta r) - \pi r^2= \pi(\Delta r)^2+2\pi r \Delta r = \pi(0.001)^2+0.002 \pi r = \pi (0.00001 + 0.002 r)$

It is a two terms expression, not three.

b. The average expression is built by the increment of area A divided by the increment of radius:

$ave(\Delta A) = \frac{A(r+ \Delta r) - A(r)}{(r+\Delta r) - r} = \frac{\Delta A}{\Delta r} = \pi \frac{ ((\Delta r)^2+2 r \Delta r ) }{\Delta r}=\pi (\Delta r + 2r)$ .

These values have to be specified for $0< \Delta r < 0.001$ .

Another condition is needed for this step: $\Delta r 0$ . Otherwise, for $\Delta r = 0$ , $\Delta A = 0$ (trivial).

c. From b., applyng $\Delta r = 0.001$ ,

$ave(\Delta A) = \pi (0.001 + 2r)$ .

d. This is expression results in a linear expression. The first term is similar to a circumference whose radius is half of 0.001; the second one is similar to the circumference's original expression, due to its second term, $2 \pi r$ .

e. This makes sense since the geometric figure formed is, in fact, a circular crown, whose width is $\Delta r$ and whose approximate lentgh is .

f. An approximate expression for the "shell" (our new circular crown) can be built by multiplying a circumference (length) by its width ( $\Delta r$ ), thus resulting in an area value.

$\Delta A* =2 \pi r \Delta r$

This is an approximation, not the exact expression that we calculated above; remember:

$\Delta A = \pi ((\Delta r)^2 + 2r \Delta r))$ .

So, we just neglected the first, smaller and negligible term, $\pi (\Delta r)^2$

g. As stated in d., $ave(\Delta A) = \pi (0.001 + 2r) = 0.001 \pi + 2 \pi r = \pi \Delta r + C(r)$ . As you can see here, the average rate of change of the area and C(r) differ by the latter expression's first term, so:

$ave(\Delta A)(r=1) = \pi (0.001 + 2) = 2.001 \pi$

Answer from: Quest

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

Answer from: Quest

the answer is: d) o, f, and kr.

atomic radius of bromine (br) is 114 pm.

atomic radius of oxygen (o) is 73 pm.

atomic radius of fluorine (f) is 72 pm.

atomic radius of krypron (kr) is 112 pm.

the atomic radius of a chemical element is a measure of the size of its atom.

the atomic radius varies with increasing atomic number, but usually increases because of increasing of number of electrons.

the atomic radius decreases across the periods because an increasing number of protons, because greater attraction between the protons and electrons.

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The area and circumference of a circle of radius r are given by a(r) = πr2 and c(r) = 2πr. suppose w...