A Quantity P Varies Partially As The Square Of M And Partially As N.When P=3.8,m=2 And N=-3.When P=-0.2,m=3 And N=2.Find(a)the Equation That Connects P,m And N,(b)the Value Of P When M=10 And N=4,(c)express M In Terms Of P And N,
what's with all the spurious capitalization?
Sie schreiben nicht auf deutsch.
p = am^2 + bn
using the given points, you now have
4a-3b = 3.8
9a+2b = -0.2
Now you can finish it off.
Post your work if you get stuck.
To find the equation that connects P, m, and N, we need to determine the relationship between these variables based on the given information.
We are told that P varies partially as the square of M and partially as N. This means that we can write the equation connecting P, M, and N as follows:
P = a * M^2 + b * N
To find the values of a and b, we need to substitute the given values of P, M, and N from the two scenarios:
For the first scenario: P = 3.8, M = 2, N = -3
3.8 = a * 2^2 + b * (-3)
For the second scenario: P = -0.2, M = 3, N = 2
-0.2 = a * 3^2 + b * 2
Now we have a system of two equations with two unknowns (a and b). We can solve this system to find the values of a and b.
Solving the system of equations:
Equation 1: 3.8 = 4a - 3b
Equation 2: -0.2 = 9a + 2b
Multiply Equation 2 by 2: -0.4 = 18a + 4b
Now we have two equations:
Equation 1: 3.8 = 4a - 3b
Equation 3: -0.4 = 18a + 4b
Adding Equation 1 and Equation 3: 3.4 = 22a
Divide both sides by 22: a = 3.4 / 22 = 0.1545
Now substitute the value of a into Equation 1: 3.8 = 4 * 0.1545 - 3b
Simplifying the equation: 3.8 = 0.618 - 3b
Rearranging the equation: 3b = 0.618 - 3.8 = -3.182
Divide both sides by 3: b = -3.182 / 3 = -1.0607
Therefore, the equation that connects P, M, and N is: P = 0.1545 * M^2 - 1.0607 * N
To find the value of P when M = 10 and N = 4, substitute these values into the equation:
P = 0.1545 * 10^2 - 1.0607 * 4
P = 0.1545 * 100 - 4.2428
P = 15.45 - 4.2428
P = 11.2072
Therefore, when M = 10 and N = 4, P = 11.2072.
To express M in terms of P and N, rearrange the equation:
P = 0.1545 * M^2 - 1.0607 * N
Move the terms with M to one side:
0.1545 * M^2 = P + 1.0607 * N
Divide both sides by 0.1545:
M^2 = (P + 1.0607 * N) / 0.1545
Take the square root of both sides:
M = sqrt((P + 1.0607 * N) / 0.1545)
Therefore, M is expressed in terms of P and N as:
M = sqrt((P + 1.0607 * N) / 0.1545)
To find the equation connecting P, m, and N, we need to understand how P varies with the square of M and with N.
Given that P varies partially as the square of M and partially as N, we can write the equation as follows:
P = kM^2 + lN
Where k and l are constants representing the respective partial variations of P with M^2 and N.
Now let's find the values of k and l using the provided data:
When P = 3.8, M = 2, and N = -3:
3.8 = k(2)^2 + l(-3)
3.8 = 4k - 3l ...(1)
When P = -0.2, M = 3, and N = 2:
-0.2 = k(3)^2 + l(2)
-0.2 = 9k + 2l ...(2)
Solving equations (1) and (2) simultaneously will give us the values of k and l.
To solve the equations, let's first multiply equation (2) by 2 to eliminate l:
-0.4 = 18k + 4l ...(3)
Adding equations (3) and (1) eliminates l:
3.4 = 22k
k = 3.4 / 22
k ≈ 0.1545
We can substitute the value of k back into equation (1) to find l:
3.8 = 4(0.1545) - 3l
3.8 ≈ 0.618 - 3l
3l ≈ 3.8 - 0.618
3l ≈ 3.182
l ≈ 3.182 / 3
l ≈ 1.061
Now that we have determined the values of k and l, the equation that connects P, M, and N is:
P = 0.1545M^2 + 1.061N
Moving on to the next part of the question:
(b) To find the value of P when M = 10 and N = 4, we can substitute these values into the equation we derived:
P = 0.1545(10)^2 + 1.061(4)
P = 0.1545(100) + 1.061(4)
P = 15.45 + 4.244
P ≈ 19.694
Therefore, when M = 10 and N = 4, P ≈ 19.694.
(c) To express M in terms of P and N, we rearrange the equation:
P = 0.1545M^2 + 1.061N
Rearranging for M:
0.1545M^2 = P - 1.061N
M^2 = (P - 1.061N) / 0.1545
M ≈ √((P - 1.061N) / 0.1545)
So, M ≈ √((P - 1.061N) / 0.1545)