Question
Find the area of A rectangles 3x^3 and -4x^3+
Answers
Answer
A : 12x^6 -15x^3
B: -12x^9+15x^3
C: -12x^6+15x^3
D: -x^3+t
B: -12x^9+15x^3
C: -12x^6+15x^3
D: -x^3+t
Answer
Find the area of a rectangle if the sides are 3x^3 and -4x^3+5
A : 12x^6 -15x^3
B: -12x^9+15x^3
C: -12x^6+15x^3
D: -x^3+50
A : 12x^6 -15x^3
B: -12x^9+15x^3
C: -12x^6+15x^3
D: -x^3+50
Answered by
GPT-4o mini
To find the area between the two curves represented by the functions \( f(x) = 3x^3 \) and \( g(x) = -4x^3 \), we first need to determine the points of intersection of these two functions. We do this by setting them equal to each other:
\[
3x^3 = -4x^3
\]
Adding \( 4x^3 \) to both sides yields:
\[
3x^3 + 4x^3 = 0
\]
\[
7x^3 = 0
\]
Dividing both sides by 7 gives:
\[
x^3 = 0 \implies x = 0
\]
This means that the two functions intersect at \( x = 0 \). Since both functions are cubic and share the same general shape (both are odd functions), they will only intersect at this point, and we need to determine the area between the curves from negative to positive limits (since cubic functions extend to negative infinity), or at some finite limits if specified.
We can choose another point to calculate the area. In particular, we can evaluate from \( x = -1 \) to \( x = 1 \).
To find the area between the curves, we take the higher curve minus the lower curve:
From \( x = -1 \) to \( x = 0 \):
- \( g(x) = -4x^3 \) is higher than \( f(x) = 3x^3 \).
- Area \( A_1 = \int_{-1}^{0} (g(x) - f(x)) \, dx \).
From \( x = 0 \) to \( x = 1 \):
- \( f(x) = 3x^3 \) is higher than \( g(x) = -4x^3 \).
- Area \( A_2 = \int_{0}^{1} (f(x) - g(x)) \, dx \).
The total area \( A \) between the curves is given by:
\[
A = A_1 + A_2
\]
Calculating \( A_1 \):
\[
A_1 = \int_{-1}^{0} \left( -4x^3 - 3x^3 \right) dx = \int_{-1}^{0} (-7x^3) \, dx = -7 \int_{-1}^{0} x^3 \, dx
\]
The integral of \( x^3 \) is \( \frac{x^4}{4} \):
\[
A_1 = -7 \left[ \frac{x^4}{4} \right]_{-1}^{0} = -7 \left[ \frac{0^4}{4} - \frac{(-1)^4}{4} \right] = -7 \left[ 0 - \frac{1}{4} \right] = -7 \left( -\frac{1}{4} \right) = \frac{7}{4}
\]
Calculating \( A_2 \):
\[
A_2 = \int_{0}^{1} \left( 3x^3 - (-4x^3) \right) dx = \int_{0}^{1} (3x^3 + 4x^3) \, dx = \int_{0}^{1} (7x^3) \, dx = 7 \int_{0}^{1} x^3 \, dx
\]
Again, using the integral of \( x^3 \):
\[
A_2 = 7 \left[ \frac{x^4}{4} \right]_{0}^{1} = 7 \left[ \frac{1^4}{4} - \frac{0^4}{4} \right] = 7 \left[ \frac{1}{4} - 0 \right] = \frac{7}{4}
\]
Finally, the total area \( A \) is:
\[
A = A_1 + A_2 = \frac{7}{4} + \frac{7}{4} = \frac{14}{4} = \frac{7}{2}
\]
Thus, the area between the rectangles represented by the curves \( 3x^3 \) and \( -4x^3 \) from \( x = -1 \) to \( x = 1 \) is \( \frac{7}{2} \) square units.
\[
3x^3 = -4x^3
\]
Adding \( 4x^3 \) to both sides yields:
\[
3x^3 + 4x^3 = 0
\]
\[
7x^3 = 0
\]
Dividing both sides by 7 gives:
\[
x^3 = 0 \implies x = 0
\]
This means that the two functions intersect at \( x = 0 \). Since both functions are cubic and share the same general shape (both are odd functions), they will only intersect at this point, and we need to determine the area between the curves from negative to positive limits (since cubic functions extend to negative infinity), or at some finite limits if specified.
We can choose another point to calculate the area. In particular, we can evaluate from \( x = -1 \) to \( x = 1 \).
To find the area between the curves, we take the higher curve minus the lower curve:
From \( x = -1 \) to \( x = 0 \):
- \( g(x) = -4x^3 \) is higher than \( f(x) = 3x^3 \).
- Area \( A_1 = \int_{-1}^{0} (g(x) - f(x)) \, dx \).
From \( x = 0 \) to \( x = 1 \):
- \( f(x) = 3x^3 \) is higher than \( g(x) = -4x^3 \).
- Area \( A_2 = \int_{0}^{1} (f(x) - g(x)) \, dx \).
The total area \( A \) between the curves is given by:
\[
A = A_1 + A_2
\]
Calculating \( A_1 \):
\[
A_1 = \int_{-1}^{0} \left( -4x^3 - 3x^3 \right) dx = \int_{-1}^{0} (-7x^3) \, dx = -7 \int_{-1}^{0} x^3 \, dx
\]
The integral of \( x^3 \) is \( \frac{x^4}{4} \):
\[
A_1 = -7 \left[ \frac{x^4}{4} \right]_{-1}^{0} = -7 \left[ \frac{0^4}{4} - \frac{(-1)^4}{4} \right] = -7 \left[ 0 - \frac{1}{4} \right] = -7 \left( -\frac{1}{4} \right) = \frac{7}{4}
\]
Calculating \( A_2 \):
\[
A_2 = \int_{0}^{1} \left( 3x^3 - (-4x^3) \right) dx = \int_{0}^{1} (3x^3 + 4x^3) \, dx = \int_{0}^{1} (7x^3) \, dx = 7 \int_{0}^{1} x^3 \, dx
\]
Again, using the integral of \( x^3 \):
\[
A_2 = 7 \left[ \frac{x^4}{4} \right]_{0}^{1} = 7 \left[ \frac{1^4}{4} - \frac{0^4}{4} \right] = 7 \left[ \frac{1}{4} - 0 \right] = \frac{7}{4}
\]
Finally, the total area \( A \) is:
\[
A = A_1 + A_2 = \frac{7}{4} + \frac{7}{4} = \frac{14}{4} = \frac{7}{2}
\]
Thus, the area between the rectangles represented by the curves \( 3x^3 \) and \( -4x^3 \) from \( x = -1 \) to \( x = 1 \) is \( \frac{7}{2} \) square units.
Answered by
GPT-4o mini
It seems you have provided a list of functions, but you haven't specified what you would like to do with them. Here are a few possible interpretations:
1. **Finding the intersection points:** If you're looking to find the points where these functions intersect, we could set pairs of functions equal to each other and solve for \( x \).
2. **Calculating the area between curves:** If you're interested in the areas between these curves over a specific interval, we could calculate integrals based on the differences between the functions.
3. **Simplifying or transforming:** If you're looking to simplify any of the expressions or transform them in a certain way, please let me know.
4. **Identifying properties:** If you need help analyzing properties such as maximums, minimums, or derivatives, I can assist with that as well.
Please clarify what you would like to do with these functions \( A, B, C, \) and \( D \).
1. **Finding the intersection points:** If you're looking to find the points where these functions intersect, we could set pairs of functions equal to each other and solve for \( x \).
2. **Calculating the area between curves:** If you're interested in the areas between these curves over a specific interval, we could calculate integrals based on the differences between the functions.
3. **Simplifying or transforming:** If you're looking to simplify any of the expressions or transform them in a certain way, please let me know.
4. **Identifying properties:** If you need help analyzing properties such as maximums, minimums, or derivatives, I can assist with that as well.
Please clarify what you would like to do with these functions \( A, B, C, \) and \( D \).
Answered by
GPT-4o mini
To find the area of a rectangle formed by two functions, we need to identify the height and the width of the rectangle. In this case, the sides of the rectangle are given as \( f(x) = 3x^3 \) and \( g(x) = -4x^3 + 5 \).
The area \( A(x) \) of the rectangle as a function of \( x \) is given by:
\[
A(x) = f(x) \cdot g(x)
\]
Plugging in the provided functions:
\[
A(x) = (3x^3)(-4x^3 + 5)
\]
Now, let's simplify this expression:
\[
A(x) = 3x^3(-4x^3) + 3x^3(5)
\]
\[
A(x) = -12x^6 + 15x^3
\]
Thus, the area of the rectangle as a function of \( x \) is:
\[
A(x) = -12x^6 + 15x^3
\]
Now, based on the options you provided:
A: \( 12x^6 - 15x^3 \)
B: \( -12x^9 + 15x^3 \)
C: \( -12x^6 + 15x^3 \)
D: \( -x^3 + 50 \)
The correct answer for the area of the rectangle formed by the sides \( 3x^3 \) and \( -4x^3 + 5 \) is:
**C: \( -12x^6 + 15x^3 \)**.
The area \( A(x) \) of the rectangle as a function of \( x \) is given by:
\[
A(x) = f(x) \cdot g(x)
\]
Plugging in the provided functions:
\[
A(x) = (3x^3)(-4x^3 + 5)
\]
Now, let's simplify this expression:
\[
A(x) = 3x^3(-4x^3) + 3x^3(5)
\]
\[
A(x) = -12x^6 + 15x^3
\]
Thus, the area of the rectangle as a function of \( x \) is:
\[
A(x) = -12x^6 + 15x^3
\]
Now, based on the options you provided:
A: \( 12x^6 - 15x^3 \)
B: \( -12x^9 + 15x^3 \)
C: \( -12x^6 + 15x^3 \)
D: \( -x^3 + 50 \)
The correct answer for the area of the rectangle formed by the sides \( 3x^3 \) and \( -4x^3 + 5 \) is:
**C: \( -12x^6 + 15x^3 \)**.
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