A couple of Achiever Math Questions

[vcuchic]

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Hey Guys,

Below are a couple of math questions that I don't understand, and hopefully some of yall do! Thank you!

1) A 120 m trapping net stretched out into a 3-sided rectangular corral is used to front a flowing river. What value of widths (w), being parallel to the riverbankds, will provide the corral with largest capturing area?

Answer: 30 m
Explanation:
Length of corral, l=120-2w
Capturing Area, A = lw = (120-2w)w = 120w-2w^2

A = -2(w^2-60w) = -2(w^2-60w+30^2-30^2) = -2(w-30)^2 + 1800

Note: I understand everything up to the red bolded part..Where are they getting the 30^2, and why is it in there.

2) Solve for x if 5x/3 + 4 = 3x - 2/5

Explanation:
5x/3 + 4 = 3x - 2/5
25x+60 = 45x - 6
20x = 66
x = 33/10

Note: How did they get from the original equation to the second step?

3) How many different lineups are possible for 4 crayons to be selected from a 7 crayon set of 7 assorted colors?

Answer: 840

Note: I know this has to do something with the permutation rule...but I don't understand how to use this rule.

4) sin^4(theta) - cos^4(theta) / sin 2(theta)

Answer: -cot2(theta)

Note: I just have no idea about this one.


Thank you so much for your time and help 😀

 

[AaBbZz]

For problem number 2 write the equation out first. Then subtract 5x/3 from 3x and add 2/5 to 4. then solve for X

For problem number 3 this is a permutation. Permutations is when order matters. Since they are asking you for different possible line-ups instead of how many combinations of crayons or groups of crayons you use the permutation formula which is N!/(N-R)! N=total crayons R=4 the number of crayons you want in the line up

For problem number 3 are you sure the answer isn't 1-cot^2theta because how I would do it write the numerator as a difference of squares

(sin2-cos2)(sin2+cos2)/sin2

Sin2+cos2=1 this is an identity so you get

sin2-cos2/sin2 which should be

1-cot2

 

[TimeforDAT]

A = -2(w^2-60w) = -2(w^2-60w+30^2-30^2) = -2(w-30)^2 + 1800


its suppose to be -2(w^2 - 60w + 30^2)...where are you getting the 1800 from?

not sure why you have that -30^2 there...

but what you are seeing is called Completing the square. you need to do that in order to solve for w, since you only have w^2 - 60w, you need to solve for that third and last term. The way to do that is take the middle coefficient 60, divide that by 2 and square it and add it to the original equation, hence you get -2(w^2-60w+30^2) which simplifies into -2(w-30)^2, w= 30

 

[Streetwolf]

#1 should be using calculus but since the DAT technically doesn't have calc on it they did that.

It's called completing the square. Since you had w^2 - 60w, you need a +30^2 or 900 to have a perfect square.

So (w - 30)^2 = w^2 - 60w + 900.

To put the 900 in you can add in the +900 - 900 terms. Since they cancel out, you haven't changed the equation at all.

Then since you only need the +900 term in there, you have to take out the -900 term. Since you have a coefficient of -2 on the outside, you take -900 * -2 and get +1800. This becomes the term after the square.

Now you look at what you have. For w = 30 you get just 1800. Since (w-30)^2 is always positive and you have a -2 as a coefficient, any other value of w will give you an area less than 1800. Therefore, w = 30 is the width that gives you the max area.

==

I would have used calculus:

Area = l*w = (120-2w)*w = 120w-2w^2.

Derivative = 120-4w.

Set derivative equal to 0 and you get w = 30.

 

[vcuchic]

geez the calculus way is so much easier
I still don't quite get the algebra way, but I think it is easier to adapt the calculus way..

so with that said, what is a derivative? and when do we use it?
Thanks for all the help!

 

[vcuchic]

For problem number 2 write the equation out first. Then subtract 5x/3 from 3x and add 2/5 to 4. then solve for X

For problem number 3 this is a permutation. Permutations is when order matters. Since they are asking you for different possible line-ups instead of how many combinations of crayons or groups of crayons you use the permutation formula which is N!/(N-R)! N=total crayons R=4 the number of crayons you want in the line up

Oh got it.
What is they ask you for just how many combinations in general(with no regard to order)?
Is there another equation we use for those type of questions?