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Daniel [21]
3 years ago
13

Find the number a such that the line x = a bisects the area under the curve y = 1/x2 for 1 ≤ x ≤ 4. 8 5​ (b) find the number b s

uch that the line y = b bisects the area in part (a).
Mathematics
1 answer:
IceJOKER [234]3 years ago
3 0

a. We're looking for a such that

\displaystyle\int_1^a\frac{\mathrm dx}{x^2}=\int_a^4\frac{\mathrm dx}{x^2}

-\dfrac1x\bigg|_{x=1}^{x=a}=-\dfrac1x\bigg|_{x=a}^{x=4}

1-\dfrac1a=\dfrac1a-\dfrac14

\dfrac54=\dfrac2a\implies\boxed{a=\dfrac85}

b. Integrating with respect to y will make things easier.

y=\dfrac1{x^2}\implies x=\dfrac1{\sqrt y}

(where we take the positive square root because we know x>0)

Now we want to find b such that

\displaystyle\int_0^b\frac{\mathrm dy}{\sqrt y}=\int_b^1\frac{\mathrm dy}{\sqrt y}

2\sqrt y\bigg|_{y=0}^{y=b}=2\sqrt y\bigg|_{y=b}^{b=1}

2\sqrt b=2-2\sqrt b

4\sqrt b=2\implies\boxed{b=\dfrac14}

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yKpoI14uk [10]

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Step-by-step explanation:

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6 0
3 years ago
Please help. Marika is training for a race.
Anika [276]

Let's call n the number of days Marika's been training for the race, and a_n the distance she runs on the nth day in meters. After the first day, when n = 1, she runs 100 meters, so


a_1=100


On the second day, she runs an additional 4 meters, on the third day, another 4, and so on. Here's what that looks like mathematically:


a_2=100 + 4\\a_3=100 + 4 + 4\\a_4=100+4+4+4


It would be easier to write this continued addition as multiplication, in which case those same equations would look like


a_2 = 100 + 4(1)\\a_3 = 100 + 4(2)\\a_4=100+4(3)


Notice that, in every case, the number 4 is being multiplied by is 1 less than n. We could even write for our first term that a_1=100+4(0). In general, we can say that


a_n=100+4(n-1)


Which is expressed by option B.


(Bonus: What piece of information from this question did we not need to use here?)

3 0
4 years ago
Add three data values to the following data set so the mean increases by 10 but the median does not change. The numbers are 42 3
777dan777 [17]
In order the numbers are 20, 29, 32, 37, 42.
So that means the median is 32. In order for the median to stay the same one of the three data values must be 32 as well as adding a number to each side of the ends. One 32 or less another 32or more.
The mean right now is: 20+29+32+37+42=160 and there are 5 values.
Mean: 160/5=32
For it to increase by 10 it has to be 42.
extra 3 numbers I used were 31, 32, & 113 (those are your answer)
3 0
3 years ago
What is the interval notation for x+1>34?
vodka [1.7K]

Explanation:

<u><em>First you subtract by -1 both sides of an equation.</em></u>

<u><em>x+1-1=34-1</em></u>

<u><em>Then, simplify the number.</em></u>

<u><em>34-1=33</em></u>

<u><em>x>33</em></u>

<u><em>Or interval notation 33,∞ </em></u>

<u><em>Final answer: → x>33 and 33,∞</em></u>

<u><em>Hope this helps!</em></u>

<u><em>Thanks!</em></u>

8 0
3 years ago
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