What is the area of the region bound by the graphs of f(x)=-x^(2),g(x)=3x-10, and x=0 in quadrant IV? Choose 1 answer: 2 (8)/(3) (C) (16)/(3) (D) (34)/(3)

Question

What is the area of the region bound by the graphs of  f(x)=-x^(2),g(x)=3x-10, and  x=0 in quadrant IV? Choose 1 answer: 2  (8)/(3) (C)  (16)/(3) (D)  (34)/(3)

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Identify Intersection Points: Identify the points of intersection between the parabola f(x) = -x^2 and the line g(x) = 3x - 10. To find the points of intersection, set f(x) equal to g(x): -x^2 = 3x - 10Solve Equation for x: Solve the equation -x^2 = 3x - 10 for x. Rearrange the equation to form a quadratic equation: x^2 + 3x - 10 = 0Factor or Use Quadratic Formula: Factor the quadratic equation or use the quadratic formula to find the roots. The quadratic factors as (x + 5)(x - 2) = 0, so the roots are x = -5 and x = 2.Determine Root in Quadrant IV: Determine which root is in quadrant IV. Since quadrant IV is where x is positive and y is negative, we only consider the root x = 2.Calculate Area of Region: Calculate the area of the region bound by the graphs and x=0. The area is a trapezoid with vertices at (0,0), (2,0), (2,-4), and (0,-10). The two bases of the trapezoid are on the y-axis and the line g(x), and the height is the distance along the x-axis from x=0 to x=2.Find Lengths of Bases: Find the lengths of the bases. The length of the base on the y-axis is the y-value of g(x) at x=0, which is g(0) = -10. The length of the base on the line g(x) is the y-value of g(x) at x=2, which is g(2) = 3(2) - 10 = -4.Calculate Height of Trapezoid: Calculate the height of the trapezoid. The height is the distance between x=0 and x=2, which is 2 units.Use Area Formula: Use the formula for the area of a trapezoid, A = (1/2)(b1 + b2)h, where b1 and b2 are the lengths of the bases and h is the height. Substitute the values into the formula: A = (1/2)(-10 + (-4))(2)Simplify to Find Area: Simplify the expression to find the area. A = (1/2)(-14)(2) A = -14

What is the area of the region bound by the graphs of $f(x)=-x^{2}, g(x)=3 x-10$ , and $x=0$ in quadrant IV? $\newline$ Choose $1$ answer: $\newline$ $2$ $\newline$ $\frac{8}{3}$ $\newline$ (C) $\frac{16}{3}$ $\newline$ (D) $\frac{34}{3}$

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What is the area of the region bound by the graphs of f(x)=−x2,g(x)=3x−10 f(x)=-x^{2}, g(x)=3 x-10 f(x)=−x2,g(x)=3x−10, and x=0 x=0 x=0 in quadrant IV?\newlineChoose 111 answer:\newline222\newline83 \frac{8}{3} 38​\newline(C) 163 \frac{16}{3} 316​\newline(D) 343 \frac{34}{3} 334​

What is the area of the region bound by the graphs of $f(x)=-x^{2}, g(x)=3 x-10$ , and $x=0$ in quadrant IV? $\newline$ Choose $1$ answer: $\newline$ $2$ $\newline$ $\frac{8}{3}$ $\newline$ (C) $\frac{16}{3}$ $\newline$ (D) $\frac{34}{3}$