Some math problem

[Dencology]

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Can you explain your answer please.
1. What is the equation of the line that is perpendicular to the line y= (-x/2) +1 and passes through the point (2,1)?

2. a boy is given a bag of candies. He eats ¼ of the candies right away. Later, he eats 1/3 of the remaining candies. If there are now 18 candies left, how many were in the bag originally?

• 36
• 40
• 48
• 60
• 72

3. if f(x) = x2 – (1/x2), for which of the following values of x does f(x) = f(2)

• -2
• -1/2
• -1/4
• ¼
• ½

4. the average of a, b, c, and d is equal to the avg. of a and d. therefore,

• b=c
• b+c=0
• a+d=b+c
• a+b=c+d
• a+c=b+d

5. what is the greatest possible value of the function f(x) = 3 cos(x+3) where x is in radians?

• 3/2
• 3
• 3pi
• 6
• 6pi

6. Let T be the sum of the circumferences of the three small congruent circles, and let C be the circumference of the large circle. What is the value of C/T?

• pi/3
• 1
• 3/pi
• 2pi/3
• 3

 

[DentalDeac]

Ok for 1. Perpendicular means you take the negative inverse of the slope. So here, our slope is -1/2 so the slope of the perpendicular line will be 2. So right now we have y= 2x + b

Plug in (2,1) for (x,y) to solve for b.

1= 2(2) + b
1= 4 + b
b= -3

so it should be y=2x - 3

 

[DentalDeac]

So for 2, the first time he eats the candies he has 3/4 left, so 3/4x. Then he eats another 1/3 of the remaining candies, leaving 2/3 of that so (2/3)*(3/4)x.

2/3 * 3/4 = 6/12 = 1/2 X

if 1/2 X = 18 then X = 36

 

[DentalDeac]

4. lets use numbers here. Say a=10 and d=30. The average of a+d = 20

If the average of a, b, c, d is equal to 20, then the sum must be 80

Therefore, knowing that a + d = 40, b+c also = 40 so I would say

a+d = b+c

 

[Mstoothlady2012]

Can you explain your answer please.

3. if f(x) = x2 – (1/x2), for which of the following values of x does f(x) = f(2) since x is squared here it won't matter if the number is negative or positive. This means that f(2) = f(-2)

• -2
• -1/2
• -1/4
• ¼
• ½

-

 

[Streetwolf]

4. lets use numbers here. Say a=10 and d=30. The average of a+d = 20

If the average of a, b, c, d is equal to 20, then the sum must be 80

Therefore, knowing that a + d = 40, b+c also = 40 so I would say

a+d = b+c

That doesn't really rule anything out. It could have been luck. Either come up with cases that eliminate the other choices or come up with a generalized case. I'm not trying to be harsh but sometimes with these problems people will pick numbers and get one lucky case where it works.

If you want the general way, it's really simple.

(a + b + c + d)/4 is the average of a, b, c, d.
(a + d)/2 is the average of a and d.

Set them equal and multiply the latter by 2/2 to put 4s in both denominators.

So you have (a + b + c + d)/4 = (2a + 2d)/4.

Since the denoms are the same, consider the nums.

a + b + c + d = 2a + 2d

Thus

b + c = a + d

And there's your answer. Solid proof.

 

[Streetwolf]

5. what is the greatest possible value of the function f(x) = 3 cos(x+3) where x is in radians?

• 3/2
• 3
• 3pi
• 6
• 6pi

3. Don't let the numbers throw you off. The cosine function oscillates between 1 and -1 so the highest value it can ever be is 1. With the 3 as the amplitude, the highest value here is 3*1 = 3.

6. Let T be the sum of the circumferences of the three small congruent circles, and let C be the circumference of the large circle. What is the value of C/T?

• pi/3
• 1
• 3/pi
• 2pi/3
• 3

If the 3 circles are congruent then they are equal in size (diameter). Each of their diameters is 1/3 the diameter of the big circle (this is taken from your picture that you posted as an attachment in another thread for all those wondering). So the sum of the circumferences of the 3 circles is (pi*d + pi*d + pi*d) = pi(d + d + d) where each d = 1/3 the diameter of the big circle. So you have pi*d as the circumference of the little circles added up. This is the same as the circumference of the big circle. The ratio is 1.

Answers in bold.

 

[250rsavage]

3. if f(x) = x2 – (1/x2), for which of the following values of x does f(x) = f(2)

• -2
• -1/2
• -1/4
• ¼
• ½

Could someone please elaborate on how #3 is worked out. I thought if f(x) = f(2) then you would just solve by putting 2 in place of x. when I do that I do get any of the answers, can someone explain?
Thanks

 

[Streetwolf]

3. if f(x) = x2 – (1/x2), for which of the following values of x does f(x) = f(2)

• -2
• -1/2
• -1/4
• ¼
• ½

Could someone please elaborate on how #3 is worked out. I thought if f(x) = f(2) then you would just solve by putting 2 in place of x. when I do that I do get any of the answers, can someone explain?
Thanks

When you have f(2) you get an answer by plugging in x = 2. The problem wants you to come up with another value of x that gives you the same answer as when you say x = 2. In this case it is -2 because every time you see an x it is squared.

You are thinking of f(x) such that x = 2. This asks for f(x) such that f(x) = f(2).

 

[250rsavage]

gotcha
thanks streetwolf