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 $x$ (since we are interested in 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, $a = 0$, $b$ is the intersection point, $f(x)=-2x^{2}-0.5x + 9$ and $g(x)=-1.75x + 7$. So $A=\int_{0}^{b}[(-2x^{2}-0.5x + 9)-(-1.75x + 7)]dx=\int_{0}^{b}(-2x^{2}+1.25x + 2)dx$.

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: Calculate with a calculator

After finding the value of $b$ from the quadratic formula and substituting it into the above expression, and then using a calculator to evaluate and round to the nearest thousandth.

Let's first solve $2x^{2}-1.25x - 2 = 0$. The quadratic formula $x=\frac{1.25\pm\sqrt{(-1.25)^{2}-4\times2\times(-2)}}{2\times2}=\frac{1.25\pm\sqrt{1.5625 + 16}}{4}=\frac{1.25\pm\sqrt{17.5625}}{4}=\frac{1.25\pm4.19076}{4}$. The positive root is $x=\frac{1.25 + 4.19076}{4}=\frac{5.44076}{4}=1.36019$.

Now, $\int_{0}^{1.36019}(-2x^{2}+1.25x + 2)dx=\left[-\frac{2}{3}x^{3}+\frac{1.25}{2}x^{2}+2x\right]_{0}^{1.36019}$.

$-\frac{2}{3}(1.36019)^{3}+\frac{1.25}{2}(1.36019)^{2}+2(1.36019)-0$

$-\frac{2}{3}(2.5277)+\frac{1.25}{2}(1.8497)+2.7204$

$-1.6851+1.1561+2.7204$

$2.1914\approx2.191$

Answer:

$2.191$