Which is closest to the area of the shaded region in the figure, in square units?
2 answers:
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Answer:
See bolded below.
Step-by-step explanation:
Take a look at the attachment below. It represents circle x and y, with respect to each of their radii. In each of the options we are given, we would have to translate and dilate the circles, by a fixed scale factor -
Now the first thing one would do is translate the circles so that they share a common center point, or in other words the center of one circle rests on the edge of the other circle. That way when dilating circle y, it may fit into circle x as it expands.
The second point is how much this smaller circle ( circle y ) has to expand. The radius of circle y being 2, has to increase by 3 times the value to equal the radius of circle x, and hence has to dilate by a scale factor of 3 as to match circle x,
<u><em>Solution = " Translate the circles so the center of one circle rests on the edge of the other circle, and dilate circle Y by a scale factor of 3 " / Option D</em></u>
To begin with, the question is asking for the answer in days, so let's change 3 weeks to days. There are 7 days in 1 week; 3 weeks times 7 days = 21 days.
21 total days of vacation.
Max + Jared + Wesley = 21 days
(Let's use the first letter of their name to represent their vacation time)
M + J + W = 21 [This is the equation we'll be coming back to]
Now, we can use the clues given in the question to have an equation for each variable/boy.
Max was on vacation twice as long as Jared. We can interpret this as M= 2J Max was on vacation only half as long as Wesley. We can interpret this as M = (1/2)W.
So far we have these three equations: M + J + W = 21M = 2J M = (1/2)W
To have an equation for each individual boy, we must rearrange the last two equations in the list.
First, M = 2J.
Divide both sides by 2
M/2 = 2J/2
(1/2)M = J
Second, <span>M = (1/2)W
Multiply both sides by 2
2M = W
New equations:
</span><span>M + J + W = 21 [From the old list]
</span>J = (1/2)W
W = 2M
Now we can substitute the last two equations into the first one.
M + J + W = 21
M + (1/2)W + 2M = 21
[Combine Like Terms]
<span>(7/2)M = 21
</span>
Then, solve for M (Max's vacation days):
Multiply both sides by 2/7
M = 6
Now we know Max was on vacation for 6 days.
If Max was on vacation twice as long as Jared, that means Jared was on vacation HALF as long as Max.
So...
<span>J = (1/2)M
J = (1/2) * 6
J = 3
Jared was on vacation for 3 days
</span>
Wesley was on vacation twice as long as Max so... W = 2M
W= 2*6
W = 12
Wesley was on vacation for 12 days.
Let's double check our answer:
M + J + W = 21 days<span>6 + 3 + 12 = 21
The numbers work out so the math is correct. Hope this helps and makes sense!</span>
21 total days of vacation.
Max + Jared + Wesley = 21 days
(Let's use the first letter of their name to represent their vacation time)
M + J + W = 21 [This is the equation we'll be coming back to]
Now, we can use the clues given in the question to have an equation for each variable/boy.
Max was on vacation twice as long as Jared. We can interpret this as M= 2J Max was on vacation only half as long as Wesley. We can interpret this as M = (1/2)W.
So far we have these three equations: M + J + W = 21M = 2J M = (1/2)W
To have an equation for each individual boy, we must rearrange the last two equations in the list.
First, M = 2J.
Divide both sides by 2
M/2 = 2J/2
(1/2)M = J
Second, <span>M = (1/2)W
Multiply both sides by 2
2M = W
New equations:
</span><span>M + J + W = 21 [From the old list]
</span>J = (1/2)W
W = 2M
Now we can substitute the last two equations into the first one.
M + J + W = 21
M + (1/2)W + 2M = 21
[Combine Like Terms]
<span>(7/2)M = 21
</span>
Then, solve for M (Max's vacation days):
Multiply both sides by 2/7
M = 6
Now we know Max was on vacation for 6 days.
If Max was on vacation twice as long as Jared, that means Jared was on vacation HALF as long as Max.
So...
<span>J = (1/2)M
J = (1/2) * 6
J = 3
Jared was on vacation for 3 days
</span>
Wesley was on vacation twice as long as Max so... W = 2M
W= 2*6
W = 12
Wesley was on vacation for 12 days.
Let's double check our answer:
M + J + W = 21 days<span>6 + 3 + 12 = 21
The numbers work out so the math is correct. Hope this helps and makes sense!</span>