1. **State the problem:** We want to find the average value of the function $$f(t) = 72 - 60\cos\left(\frac{\pi}{3}t\right)$$ over the interval $$[0,8]$$. 2. **Formula for average value of a function:** The average value $$f_{avg}$$ of a continuous function $$f(t)$$ on $$[a,b]$$ is given by $$ f_{avg} = \frac{1}{b - a} \int_a^b f(t) \, dt $$ 3. **Apply the formula:** Here, $$a=0$$ and $$b=8$$, so $$ f_{avg} = \frac{1}{8 - 0} \int_0^8 \left(72 - 60\cos\left(\frac{\pi}{3}t\right)\right) dt = \frac{1}{8} \int_0^8 \left(72 - 60\cos\left(\frac{\pi}{3}t\right)\right) dt $$ 4. **Split the integral:** $$ \int_0^8 \left(72 - 60\cos\left(\frac{\pi}{3}t\right)\right) dt = \int_0^8 72 \, dt - 60 \int_0^8 \cos\left(\frac{\pi}{3}t\right) dt $$ 5. **Integrate each term:** - $$\int_0^8 72 \, dt = 72t \Big|_0^8 = 72 \times 8 = 576$$ - For $$\int_0^8 \cos\left(\frac{\pi}{3}t\right) dt$$, use substitution: Let $$u = \frac{\pi}{3}t$$, so $$dt = \frac{3}{\pi} du$$. When $$t=0$$, $$u=0$$; when $$t=8$$, $$u=\frac{8\pi}{3}$$. Thus, $$ \int_0^8 \cos\left(\frac{\pi}{3}t\right) dt = \int_0^{\frac{8\pi}{3}} \cos(u) \cdot \frac{3}{\pi} du = \frac{3}{\pi} \int_0^{\frac{8\pi}{3}} \cos(u) du $$ 6. **Evaluate the cosine integral:** $$ \int_0^{\frac{8\pi}{3}} \cos(u) du = \sin(u) \Big|_0^{\frac{8\pi}{3}} = \sin\left(\frac{8\pi}{3}\right) - \sin(0) = \sin\left(\frac{8\pi}{3}\right) $$ Since $$\frac{8\pi}{3} = 2\pi + \frac{2\pi}{3}$$ and sine is periodic with period $$2\pi$$, $$ \sin\left(\frac{8\pi}{3}\right) = \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} $$ 7. **Substitute back:** $$ \int_0^8 \cos\left(\frac{\pi}{3}t\right) dt = \frac{3}{\pi} \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2\pi} $$ 8. **Calculate the full integral:** $$ \int_0^8 \left(72 - 60\cos\left(\frac{\pi}{3}t\right)\right) dt = 576 - 60 \times \frac{3\sqrt{3}}{2\pi} = 576 - \frac{180\sqrt{3}}{2\pi} = 576 - \frac{90\sqrt{3}}{\pi} $$ 9. **Calculate the average value:** $$ f_{avg} = \frac{1}{8} \left(576 - \frac{90\sqrt{3}}{\pi}\right) = 72 - \frac{90\sqrt{3}}{8\pi} $$ **Final answer:** $$ \boxed{72 - \frac{90\sqrt{3}}{8\pi}} \, \approx \, 72 - 6.19 = 65.81 $$ This is the average value of the function over the interval $$[0,8]$$.