To find the container's cross-sectional area, we'll need to divide the volume polynomial by the height polynomial.
The volume \( V \) of the shipping container is given by:
\[ V = 24x^3 + 6x^2 - 9x \]
The height \( h \) of the shipping container is given as:
\[ h = 3x \]
To find the cross-sectional area \( A \), we perform polynomial division by dividing the volume \( V \) by the height \( h \):
\[ A = \frac{V}{h} = \frac{24x^3 + 6x^2 - 9x}{3x} \]
Now, let's perform the division step-by-step:
-
Divide the leading term of the numerator \( 24x^3 \) by the leading term of the denominator \( 3x \): \[ \frac{24x^3}{3x} = 8x^2 \]
-
Multiply the entire denominator by \( 8x^2 \): \[ 8x^2 \cdot (3x) = 24x^3 \]
-
Subtract this from the original volume: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x \]
-
Next, divide the leading term \( 6x^2 \) by \( 3x \): \[ \frac{6x^2}{3x} = 2x \]
-
Multiply the entire denominator by \( 2x \): \[ 2x \cdot (3x) = 6x^2 \]
-
Subtract again: \[ (6x^2 - 9x) - 6x^2 = -9x \]
-
Now, divide the leading term \( -9x \) by \( 3x \): \[ \frac{-9x}{3x} = -3 \]
-
Multiply the entire denominator by \( -3 \): \[ -3 \cdot (3x) = -9x \]
-
Subtracting: \[ (-9x) - (-9x) = 0 \]
So the division is complete, and the quotient gives us the cross-sectional area \( A \):
\[ A = 8x^2 + 2x - 3 \]
Thus, the cross-sectional area of the container is:
\[ \text{cross-sectional area} = 8x^2 + 2x - 3 \]
1 answer