<span>(1 km x 1,000m) </span>÷ 850 m/s = 1.18 s
Answer: Choice B) 1/3, 1/4, 1/5, 1/6
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Plug in n = 1 to find the first term
a_n = ( (n+1)! )/( (n+2)! )
a_1 = ( (1+1)! )/( (1+2)! )
a_1 = ( 2! )/( 3! )
a_1 = ( 2*1 )/( 3*2*1 )
a_1 = 2/6
a_1 = 1/3
The first term is 1/3. Optionally you can stop here because only choice B has 1/3 listed as the first term, so this must be the answer. However, I'm going to keep going to show how to find the three other terms. This will help confirm why choice B is the answer, and it will be handy for those times when you aren't given multiple choice answers.
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Plug in n = 2
a_n = ( (n+1)! )/( (n+2)! )
a_2 = ( (2+1)! )/( (2+2)! )
a_2 = ( 3! )/( 4! )
a_2 = ( 3*2*1 )/( 4*3*2*1 )
a_2 = 6/24
a_2 = 1/4
The second term is 1/4
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Plug in n = 3
a_n = ( (n+1)! )/( (n+2)! )
a_3 = ( (3+1)! )/( (3+2)! )
a_3 = ( 4! )/( 5! )
a_3 = ( 4*3*2*1 )/( 5*4*3*2*1 )
a_3 = 24/120
a_3 = 1/5
The third term is 1/5
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Plug in n = 4
a_n = ( (n+1)! )/( (n+2)! )
a_4 = ( (4+1)! )/( (4+2)! )
a_4 = ( 5! )/( 6! )
a_4 = ( 5*4*3*2*1 )/( 6*5*4*3*2*1 )
a_4 = 120/720
a_4 = 1/6
The fourth term is 1/6
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The first four terms are: 1/3, 1/4, 1/5, 1/6, so that confirms why choice B is the answer.
This problem is about adding up the surface area of all of the surfaces he will paint and then dividing it by 200 to find how many gallons he will need to buy. Since he won't paint his roof, we need to find the surface area of every square foot that he will paint. So to do this it is just doing the area formula A=L*W for rectangles, and A=1/2*H*W, for the triangles. So he will paint 2 triangular spaces, the front and the back, plus he will paint 4 rectangular walls. So let's do the very front section first. When looking straight at the front u see a rectangle and a triangle. So let's find the area of both. The rectangle we need to plug in the length, or in other words the height, which is 8 feet. The width is 12 ft. So what is 8 times 12? It is 96 sq. ft. So 96 square feet is the front rectangle. There is another rectangle on the back of the shed that's the same dimensions that he will paint, so instead of doing that again we can just multiply 96 sq ft by 2, which is 192 sq ft. Now let's do the 2 triangles he will paint. Both of the triangles are the same exact dimensions, so what we do is plug the numbers in, the height is 4 ft. So it looks like 1/2 * 4 ft * 12 ft. 1/2 of 4 is 2, so multiply 2 by 12, which equals 24. So one of the triangles has 24 sq ft, but since there is another one on the back side that he will paint, we multiply that by 2 to save time. This will equal 48 sq ft. Now all we have to do is the other two walls on the sides. This is the same formula, A= L*W. so length is the same as the other 2 rectangles because they are the same height, so the length is 8 ft. And the width is 20 ft. What is 20 times 8? 20 times 8 is 160 sq ft. Since there is another wall just like that on the other side, we multiply that by 2 to get 320 sq ft. Now we add all of it up. 320+ 48+ 192= 560 sq ft. So he will paint 560 sq ft. What is 560 divided by 200? It will be 2.8, so he will have to buy 2.8 gallons of paint. Since you can't buy 2.8 gallons of paint, he will really buy 3 gallons, so he will have some left over. Since 1 gallon cost $48, we will need to multiply 48 by 3, to give us 144. So he will spend $144 on paint to paint his shed. I hope this helps
However because you can use however as “but”
Her area of study is too narrow. There are other sources of water in the city that she is not considering. So her sample is too biased leaning toward the Uncle's pond water, and that leaves out every other source. We say that the Uncle's source of water is over-represented while the other sources are completely under-represented.
Let's put it this way: We have a hypothetical city that has 100 acres of pond water either in or surrounding the city limits. If the Uncle's pond only represents 1% of this, then she's ignoring the other 99% of the ponds.
What Lila needs to do to to fix this error is to draw up a map of each major pond and mark those ponds with numbers 001,002,...998,999. In this example, there are 999 ponds. Then she needs to either use a random number table or computer software to help randomly generate values to help select the ponds. Doing so will ensure that she spans a good portion of the city and not stay focused on just the narrow area of her Uncle's backyard.
The reason why she needs to enlarge her area of study is because the results of her Uncle's pond study may lead to the wrong conclusion of the city overall. Let's say his pond is contaminated somehow, and it's only his pond that's the unfortunate one. She would likely see the pond is contaminated and conclude that the whole city's water is ruined as well, which isn't the case. Or we could have nearly the entire city's pond water in trouble, but her Uncle's pond is one of the lucky ponds not to get contaminated. We can see that Lila would likely conclude that no action needs to be taken to clean up the city's water sources, which is also not the case.
If Lila only cared about her Uncle's pond, then that pond (and perhaps immediate close surrounding area) would be her population of study. However, her study is about citywide pond water which is why she needs to extend to other places in the city or in the outer surrounding areas.
