M varies jointly as N and the cube of P. When N=3 and P=-2, M=-48. given the consent of the variation find M when N=-4 and p=5

The given problem states that \( M \) varies jointly as \( N \) and the cube of \( P \). This can be expressed as:

\[
M = k \cdot N \cdot P^3
\]

where \( k \) is the constant of variation.

### Step 1: Find the constant \( k \)
We are given that when \( N = 3 \), \( P = -2 \), and \( M = -48 \). Substituting these values into the equation:

\[
-48 = k \cdot 3 \cdot (-2)^3
\]

Since \( (-2)^3 = -8 \), the equation becomes:

\[
-48 = k \cdot 3 \cdot (-8)
\]

\[
-48 = -24k
\]

Now, solve for \( k \):

\[
k = \frac{-48}{-24} = 2
\]

### Step 2: Use \( k \) to find \( M \) when \( N = -4 \) and \( P = 5 \)

Now that we know \( k = 2 \), we can use the formula to find \( M \) when \( N = -4 \) and \( P = 5 \):

\[
M = 2 \cdot (-4) \cdot (5)^3
\]

Since \( 5^3 = 125 \), the equation becomes:

\[
M = 2 \cdot (-4) \cdot 125 = 2 \cdot (-500) = -1000
\]

### Final Answer:
When \( N = -4 \) and \( P = 5 \), \( M = -1000 \).

Would you like further details or have any questions? Here are 5 related questions for practice:

1. If \( M \) varies jointly as \( N \) and \( P^2 \), and \( M = 24 \) when \( N = 2 \) and \( P = 3 \), what is the value of \( M \) when \( N = 5 \) and \( P = 4 \)?
2. How do you find the constant of variation in a joint variation problem?
3. What would happen to \( M \) if \( P \) is doubled, while \( N \) stays the same?
4. If \( M \) varies inversely as \( N \), and directly as the cube of \( P \), how would the equation change?
5. How does joint variation differ from direct variation?

**Tip:** Always double-check your exponent calculations when working with powers in joint variation problems!

M varies jointly as N and the cube of P. When N=3 and P=-2, M=-48. Given the constant of variation, find M when N=-4 and P=5.

Explore how to solve a joint variation problem where M varies jointly as N and the cube of P. This problem demonstrates key algebraic concepts, including joint variation and exponent rules. Learn how to find the constant of variation and solve for M when N=-4 and P=5. Suitable for Grades 9-11.

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