Find the A) Constant of Variation, B) the Equation of the Variation, and the C) the Solution
Suppose B varies directly with C and inversely with D and the cube root of A.
B = 4 when A = 27, C = 2, and D = 5. Find B when C = 3, A = 64, and D = 6.
To find the constant of variation, equation of variation, and the solution, we need to identify the relationships between the variables given in the problem.
Suppose B varies directly with C, inversely with D, and the cube root of A. This means that B is directly proportional to C, inversely proportional to D, and inversely proportional to the cube root of A.
Now, let's break down the problem into separate parts:
A) Constant of Variation:
To find the constant of variation between B and the variables, we need to express the relationships mathematically. In this case, we have:
B ∝ C
B ∝ 1/D
B ∝ 1/∛A
Since B is directly proportional to C and inversely proportional to D and the cube root of A, we can combine these relationships:
B ∝ C / (D * ∛A)
Now, we need to find the constant of variation (k) that makes this equation true. We can do this by substituting the known values for B, C, A, and D and solving for k:
4 = 2 / (5 * ∛27)
4 = 2 / (5 * 3)
4 = 2 / 15
To simplify the equation, we can cross-multiply:
4 * 15 = 2
60 = 2
Dividing both sides by 2, we find:
k = 60 / 2
k = 30
Therefore, the constant of variation (k) is 30.
B) Equation of Variation:
Now that we have determined the constant of variation (k), we can write the equation of variation:
B = k * (C / (D * ∛A))
Substituting the value of k:
B = 30 * (C / (D * ∛A))
C) Solution:
To find the value of B when C = 3, A = 64, and D = 6, we can plug these values into the equation of variation:
B = 30 * (3 / (6 * ∛64))
Simplifying further:
B = 30 * (3 / (6 * 4))
B = 30 * (3 / 24)
B = 30 * (1/8)
B = 3.75
Therefore, when C = 3, A = 64, and D = 6, the value of B is 3.75.
To find the constant of variation and the equation of the variation, let's break down the given information into separate variations.
A) Direct Variation: We know that B varies directly with C. In direct variation, the equation is of the form B = kC, where k represents the constant of variation. Let's find k using the given information.
When B = 4, A = 27, C = 2, and D = 5:
4 = k * 2
Solving for k:
k = 4/2
k = 2
Therefore, the constant of variation is 2.
B) Inverse Variation: We know that B varies inversely with D and the cube root of A. In inverse variation, the equation is of the form B = k/(D * ∛A). Let's find k using the given information.
When B = 4, A = 27, C = 2, and D = 5:
4 = k/(5 * ∛27)
Simplifying, since ∛27 = 3:
4 = k/(5 * 3)
Solving for k:
k = 4 * 5 * 3
k = 60
Therefore, the constant of variation for inverse variation is 60.
C) Solution: Now that we have the constant of variation for each type of variation, we can find B when C = 3, A = 64, and D = 6.
Using the direct variation equation, B = kC:
B = 2 * 3
B = 6
Using the inverse variation equation, B = k/(D * ∛A):
B = 60/(6 * ∛64)
Simplifying, since ∛64 = 4:
B = 60/(6 * 4)
B = 60/24
B = 2.5
Therefore, when C = 3, A = 64, and D = 6, B is equal to 6 using direct variation and approximately equal to 2.5 using inverse variation.