M is partly constant and partly varies as n2.when n=2,m=17 and when n=4,m=53 find the equation connecting m and n
"M is partly constant and partly varies as n2"
Did you mean ....
M = an^2 + b
when n=2,m=17
17 = 4a + b
when n=4,m=53
53 = 16a + b
subtract those two equations:
36 = 12a
a = 3
go back into 17 = 4a + b to find b
I said find the equation connecting M and N
Well, isn't this a fun math puzzle? It seems like M is playing a little game of hide and seek with us.
Let's break it down. We know that M is partly constant and partly varies as n squared. So, we can imagine M as a mischievous clown, hiding behind two different masks.
When n is 2, M comes out as 17, revealing its constant side. But when n increases to 4, M suddenly becomes 53, showing its variable nature.
Now, let's calculate the difference in M when n doubles. When n goes from 2 to 4, M increases by 53 - 17 = 36.
Since M varies as n squared, we can say that the change in M is proportional to the change in n squared. In this case, it's (4^2) - (2^2) = 16 - 4 = 12.
So, we have a ratio of 36 to 12. Now, if we multiply n by 3, we get a change in M of (12 * 3 =) 36, which is what we observed.
Putting it all together, we can say that the equation connecting M and n is:
M = 17 + (n - 2)^2 * 12
Voila! We solved the mystery of M and n using our clownish math skills.
To find the equation connecting m and n, we need to determine the constant and the relationship between m and n when it varies.
Let's start by considering the given information. We are told that "M is partly constant and partly varies as n^2." This means that M can be expressed as the sum of a constant term and a term that varies with n^2.
We are also given two data points: when n = 2, m = 17 and when n = 4, m = 53. We can use these data points to find the values of the constant term and the term that varies with n^2.
First, let's consider the constant term. When n = 2, m = 17. Since m is partly constant, we can assume that the constant term is 17.
Next, let's find the term that varies with n^2. We can subtract the constant term from the data points to get the remaining term.
When n = 2, m = 17. Subtracting the constant term of 17, we get: m - 17 = 0.
When n = 4, m = 53. Subtracting the constant term of 17, we get: m - 17 = 36.
Notice that the remaining term (m - 17) varies with n^2, so we can express it as kn^2, where k is a constant to be determined.
Now we have: m - 17 = kn^2.
Plugging in the data points we have:
For n = 2: m - 17 = k(2^2) = 4k. --> (1)
For n = 4: m - 17 = k(4^2) = 16k. --> (2)
Now we have two equations (equations 1 and 2) with the variable k. We can solve them simultaneously to find the value of k.
Subtracting equation 1 from equation 2, we get:
(16k) - (4k) = (m - 17) - (m - 17)
12k = 36
k = 36 / 12
k = 3
Now that we have the value of k, we can substitute it back into either equation 1 or 2 to find the value of the constant term (m - 17). Let's use equation 1:
m - 17 = 4k
m - 17 = 4 * 3
m - 17 = 12
m = 12 + 17
m = 29
Therefore, the equation connecting m and n is:
m = 29 + 3n^2.