Hey, we recently started a topic in class that I don't understand well, I just want to post 6 of the problems and ask if they're right or not (some of them I have no idea how to do).
1. If a varies directly as b and b = 18 when a = 27, find a when b = 10.
For this I did 27/18 = a/10 and my answer was a = 15.
2. If y varies inversely as x and y = -3 when x = 9, find y when x = 81.
For this I did -3/81= = y/9 and my answer was y = -1/3.
3. If y varies jointly as x and z, and y = 18 when x = 6 and z = 15, find y when x = 12 and z = 4.
For this I did 18/12*4 = y/6*15 and my answer was y = 33.75.
Direction variation can be expressed in the form of y = kx and inverse variation can be expressed in the form of y = k/x, right?
Last off is the part I didn't understand.
Is the relationship between the variables in set of data a direct variation, inverse variation, or neither? Indicate the constant of variation, if applicable.
a. {(0.5, 1.5), (2, 6), (6, 18)} I said this is direct variation though no idea how to get the constant of variation.
b. {(0.2, 12), (0.6, 4), (1.2, 2)} I said this was inverse variation but yet again no idea how to get the constant of variation.
c. {(1, 2), (2, 1), (3, 0.5)} I said this was neither, and I expect there is no constant of variation?
I do understand this is quite long, for whoever and if someone even does help me with this, thank you!
you did your first 3 questions by using ratios.
This is fine as long as you know where to put the numbers.
since you appear to understand that
direct variation ---> y = kx
inverse variation ---> y = k/x
perhaps that is the approach you might take at the beginning.
1. It says direct variation, so a = kb
we are told, a=27, and b = 18, so
27 = 18k
k = 27/18 = 3/2
so we know a = (3/2)b
now if b = 10
a = (3/2)(10) = 15
2. is correct
3. Using the k method:
y = k(xz)
given: x=6, y=18, z=15
18 = k(90) ---> k = 1/5
so y = (1/5)xz
when x=12, z=4 , then
y = (1/5)(12)(4) = 48/5
Your last part:
recall that for direct variation: y = kx, so k = y/x
now look at the ordered pairs, is
18/6 = 6/2 = 1.5/.5 ??? yes, all 3 are equal to 3, so it is a direct variation
for in inverse we have y = k/x ---> k = xy
look at your given points
In all cases, xy = 2.4 , so your answer of inverse is correct
for c) points are (1,2), (2,1), (3, .5)
1x2 = 2x1 ≠ 3x.5, so it is not inverse variation
2/1 ≠ 1/2 , no point going further, not direct
so neither is the correct answer
I'll be happy to help you with these problems! Let's go through each one and explain how to solve them.
1. If a varies directly as b and b = 18 when a = 27, find a when b = 10.
To solve this problem, you need to use the concept of direct variation. Direct variation can be expressed as the equation y = kx, where y and x are variables and k is the constant of variation.
Given that a varies directly as b, we can write the equation as a = kb. We are given that when b = 18, a = 27. So we can substitute these values into the equation to find k:
27 = k * 18
Dividing both sides by 18 gives us:
k = 27 / 18 = 3/2
Now, we can use the value of k to find a when b = 10:
a = (3/2) * 10 = 30/2 = 15
So your answer, a = 15, is correct!
2. If y varies inversely as x and y = -3 when x = 9, find y when x = 81.
Inverse variation can be expressed as the equation y = k/x, where y and x are variables and k is the constant of variation.
Given that y varies inversely as x, we can write the equation as y = k/x. We are given that when x = 9, y = -3. So we can substitute these values into the equation to find k:
-3 = k / 9
Multiplying both sides by 9 gives us:
k = -3 * 9 = -27
Now, we can use the value of k to find y when x = 81:
y = (-27) / 81 = -1/3
So your answer, y = -1/3, is correct!
3. If y varies jointly as x and z, and y = 18 when x = 6 and z = 15, find y when x = 12 and z = 4.
Joint variation can be expressed as the equation y = kxz, where y, x, and z are variables and k is the constant of variation.
Given that y varies jointly as x and z, we can write the equation as y = kxz. We are given that when x = 6 and z = 15, y = 18. So we can substitute these values into the equation to find k:
18 = k * 6 * 15
Dividing both sides by 6 * 15 gives us:
k = 18 / (6 * 15) = 18 / 90 = 1/5
Now, we can use the value of k to find y when x = 12 and z = 4:
y = (1/5) * 12 * 4 = 48/5 = 9.6
So your answer, y = 9.6, is correct!
For the last part, determining whether the relationship is a direct variation, inverse variation, or neither, and finding the constant of variation, if applicable, you need to analyze the given data.
a. {(0.5, 1.5), (2, 6), (6, 18)}
To determine if this is a direct variation, we need to check if the ratios of corresponding values of y and x are always the same. Let's calculate the ratios:
1.5 / 0.5 = 3
6 / 2 = 3
18 / 6 = 3
Since the ratios are all the same (3 in this case), we can conclude that this is a direct variation. The constant of variation, k, is equal to this ratio, which is 3.
b. {(0.2, 12), (0.6, 4), (1.2, 2)}
To determine if this is an inverse variation, we need to check if the products of corresponding values of y and x are always the same. Let's calculate the products:
0.2 * 12 = 2.4
0.6 * 4 = 2.4
1.2 * 2 = 2.4
Since the products are all the same (2.4 in this case), we can conclude that this is an inverse variation. The constant of variation, k, is equal to this product, which is 2.4.
c. {(1, 2), (2, 1), (3, 0.5)}
In this case, the ratios and products of corresponding values are not the same. Therefore, this set of data does not represent either direct or inverse variation, and there is no constant of variation.
I hope this helps! Feel free to ask if you have any further questions.