m is partly constant and partly varies as n2 when n = 2 m = 17 and when n = 4 and m = 53 ( a ) find the equation connecting m and n ( b ) find n when m = 32
Assuming by n2 you mean n^2, you would have
m = c + bn^2 , where c and b are constants
when n=2, m=17
17 = c + 4b
when n = 4, m = 53
53 = c + 16b
subtract those two equations:
36 = 12b
b = 3
now you can find c, and your complete equation,
then replace m with 32 and you can find n
To find the equation connecting m and n, we can use the method of substitution.
(a) Let's start by finding the relationship between m and n based on the given information.
Given:
When n = 2, m = 17
When n = 4, m = 53
Let's use these two data points to form two equations and solve for the relationship between m and n.
First equation:
m = a * n^2
Substituting the given values:
17 = a * 2^2
17 = a * 4
17/4 = a
Second equation:
m = a * n^2
Substituting the given values:
53 = a * 4^2
53 = a * 16
53/16 = a
Since both equations equal a, we can set them equal to each other:
17/4 = 53/16
Cross multiplying:
16 * 17 = 4 * 53
272 = 212
This is not a true statement, so there seems to be an error in the given information. Please double-check the values and try again.
(b) Without the correct equation connecting m and n, we cannot find the value of n when m = 32.
To find the equation connecting m and n, let's analyze the given information:
- When n = 2, m = 17
- When n = 4, m = 53
(a) To find the equation connecting m and n, one way is to assume a relationship between m and n, and then use the given data to solve for the relationship. Since m is partly constant and partly varies as n^2, we can assume the equation to be of the form:
m = a + b * n^2
where 'a' is the constant term and 'b' is the coefficient of n^2. To determine the values of a and b, we'll substitute the given data points into this equation:
When n = 2, m = 17:
17 = a + b * 2^2
17 = a + 4b
When n = 4, m = 53:
53 = a + b * 4^2
53 = a + 16b
We now have a system of equations with two variables (a and b). Subtracting the first equation from the second equation cancels out the 'a' term:
53 - 17 = (a + 16b) - (a + 4b)
36 = 16b - 4b
36 = 12b
b = 36/12
b = 3
Now, substitute the value of b back into the first equation to solve for 'a':
17 = a + 4 * 3
17 = a + 12
a = 17 - 12
a = 5
Therefore, the equation connecting m and n is: m = 5 + 3n^2.
(b) To find n when m = 32, substitute m = 32 into the equation m = 5 + 3n^2 and solve for n:
32 = 5 + 3n^2
27 = 3n^2
n^2 = 27/3
n^2 = 9
n = ±√9
Therefore, when m = 32, n can be either +3 or -3.