the function f(x) = 5/3 x - 5 is graphed below. use geometric formulas to evaluate the following definite…

Explanation:

Step1: Find function values at endpoints

When (x = 4), (f(4)=\frac{5}{3}\times4 - 5=\frac{20}{3}-5=\frac{20 - 15}{3}=\frac{5}{3}). When (x = 6), (f(6)=\frac{5}{3}\times6-5 = 10 - 5=5).

Step2: Recognize geometric shape

The definite - integral (\int_{4}^{6}(\frac{5}{3}x - 5)dx) represents the area between the line (y=\frac{5}{3}x - 5), the (x) - axis, (x = 4), and (x = 6). This area is a trapezoid.

Step3: Apply trapezoid area formula

The area formula for a trapezoid is (A=\frac{1}{2}(b_1 + b_2)h), where (b_1) and (b_2) are the lengths of the parallel sides and (h) is the height. Here, (b_1=\frac{5}{3}), (b_2 = 5), and (h=6 - 4 = 2). [ \begin{align*} A&=\frac{1}{2}(\frac{5}{3}+5)\times2\ &=\frac{1}{2}(\frac{5 + 15}{3})\times2\ &=\frac{1}{2}\times\frac{20}{3}\times2\ &=\frac{20}{3} \end{align*} ]

Answer:

(\frac{20}{3})