question the shaded region shown below is bounded by the functions f(x)=-2x² - 0.5x + 9 and g(x)=-1.75x + 7…

Explanation:

Step1: Find intersection point

Set $f(x)=g(x)$, so $-2x^{2}-0.5x + 9=-1.75x + 7$. Rearrange to $-2x^{2}+1.25x + 2 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a=-2$, $b = 1.25$, $c = 2$. We get the positive - valued intersection point relevant to the region (since we are considering the region starting from $x = 0$).

Step2: Set up integral for area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$ is given by $A=\int_{a}^{b}[f(x)-g(x)]dx$. Here, $a = 0$ and $b$ is the intersection - point value. $A=\int_{0}^{b}[(-2x^{2}-0.5x + 9)-(-1.75x + 7)]dx=\int_{0}^{b}(-2x^{2}+1.25x + 2)dx$.

Step3: Evaluate integral

Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int_{0}^{b}(-2x^{2}+1.25x + 2)dx=\left[-2\times\frac{x^{3}}{3}+1.25\times\frac{x^{2}}{2}+2x\right]_{0}^{b}$.

Step4: Use calculator

After finding the value of $b$ from the quadratic formula and substituting it into the above - evaluated integral, and using a calculator to compute the result and rounding to the nearest thousandth.

Let's first solve $-2x^{2}+1.25x + 2 = 0$. The quadratic formula $x=\frac{-1.25\pm\sqrt{(1.25)^{2}-4\times(-2)\times2}}{2\times(-2)}=\frac{-1.25\pm\sqrt{1.5625 + 16}}{-4}=\frac{-1.25\pm\sqrt{17.5625}}{-4}$. The positive root is $x=\frac{-1.25+\sqrt{17.5625}}{-4}\approx1.25$.

Now, $\int_{0}^{1.25}(-2x^{2}+1.25x + 2)dx=\left[-\frac{2}{3}x^{3}+\frac{1.25}{2}x^{2}+2x\right]_{0}^{1.25}$

$=-\frac{2}{3}(1.25)^{3}+\frac{1.25}{2}(1.25)^{2}+2(1.25)$

$=-\frac{2}{3}\times1.953125+\frac{1.25}{2}\times1.5625 + 2.5$

$=-1.302083+0.9765625 + 2.5$

$\approx2.175$

Answer:

$2.175$