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Answer :
To determine the area of the trapezoid formed by the parabola [tex]\( y = -2x^2 + 24x - 40 \)[/tex], we'll follow these steps:
1. Find the points of intersection with the x-axis:
The parabola intersects the x-axis when [tex]\( y = 0 \)[/tex]. So, we solve the equation:
[tex]\[
-2x^2 + 24x - 40 = 0
\][/tex]
Solving this quadratic equation, we find the x-values where the parabola crosses the x-axis, which are [tex]\( x = 2 \)[/tex] and [tex]\( x = 10 \)[/tex]. These points are [tex]\((2, 0)\)[/tex] and [tex]\((10, 0)\)[/tex].
2. Identify the top base of the trapezoid:
The height of 24 given in the problem represents the y-coordinate of the topmost points of the trapezoid. Since these points have the same y-coordinate, the top base of the trapezoid is parallel to the x-axis. However, in this case, the given height is the perpendicular distance (height) from the x-axis to the top of the trapezoid.
3. Calculate the bases and the height of the trapezoid:
- The length of the bottom base (on the x-axis) is the distance between the x-intercepts: [tex]\( \text{base}_1 = 10 - 2 = 8 \)[/tex] units.
- The top base of the trapezoid is effectively zero because the described y-value (24) doesn't form another base parallel to the x-axis (unless the question intended it to), but for our calculation, we'll focus on height.
- The height of the trapezoid is the y-coordinate difference between the x-axis and the given height: [tex]\( \text{height} = 24 \)[/tex].
4. Calculate the area of the trapezoid:
The area [tex]\( A \)[/tex] of a trapezoid is given by:
[tex]\[
A = \frac{\text{base}_1 + \text{base}_2}{2} \times \text{height}
\][/tex]
Substituting the values:
[tex]\[
A = \frac{8 + 0}{2} \times 24 = \frac{8}{2} \times 24 = 4 \times 24 = 96 \text{ square units}
\][/tex]
Therefore, the area of the trapezoid is 96 square units.
1. Find the points of intersection with the x-axis:
The parabola intersects the x-axis when [tex]\( y = 0 \)[/tex]. So, we solve the equation:
[tex]\[
-2x^2 + 24x - 40 = 0
\][/tex]
Solving this quadratic equation, we find the x-values where the parabola crosses the x-axis, which are [tex]\( x = 2 \)[/tex] and [tex]\( x = 10 \)[/tex]. These points are [tex]\((2, 0)\)[/tex] and [tex]\((10, 0)\)[/tex].
2. Identify the top base of the trapezoid:
The height of 24 given in the problem represents the y-coordinate of the topmost points of the trapezoid. Since these points have the same y-coordinate, the top base of the trapezoid is parallel to the x-axis. However, in this case, the given height is the perpendicular distance (height) from the x-axis to the top of the trapezoid.
3. Calculate the bases and the height of the trapezoid:
- The length of the bottom base (on the x-axis) is the distance between the x-intercepts: [tex]\( \text{base}_1 = 10 - 2 = 8 \)[/tex] units.
- The top base of the trapezoid is effectively zero because the described y-value (24) doesn't form another base parallel to the x-axis (unless the question intended it to), but for our calculation, we'll focus on height.
- The height of the trapezoid is the y-coordinate difference between the x-axis and the given height: [tex]\( \text{height} = 24 \)[/tex].
4. Calculate the area of the trapezoid:
The area [tex]\( A \)[/tex] of a trapezoid is given by:
[tex]\[
A = \frac{\text{base}_1 + \text{base}_2}{2} \times \text{height}
\][/tex]
Substituting the values:
[tex]\[
A = \frac{8 + 0}{2} \times 24 = \frac{8}{2} \times 24 = 4 \times 24 = 96 \text{ square units}
\][/tex]
Therefore, the area of the trapezoid is 96 square units.
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