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.