By Cauchy Schwarz,
(7 + 24 + 7)(7 cos t + 24 sin t + 7) <= (7^2 + 24^2 + 7^2),
so 7 cos t + 24 sin t + 7 <= 337/19.
The maximum value is 337/19.
+397 Here's a non calculus approach.
Begin by noticing that
\(7^{2}+24^{2}=49+576=625=25^{2},\)
so 7, 24, 25 can be used to as the sides of a right-angled triangle.
Let the angle opposite the 7 be \(\phi,\)
then
\(\sin\phi=7/25 \text{ and } \cos\phi=24/25.\)
\(\displaystyle 7\cos\theta+24\sin\theta+7\\ = 25(\frac{7}{25}\cos\theta+\frac{24}{25}\sin\theta)+7\\ =25(\sin\phi\cos\theta+\cos\phi\sin\theta)+7 \\ =25\sin(\theta+\phi)+7.\)
Maximum value will be 25 + 7 = 32 occuring when
\(\sin(\theta+\phi)=1, \\ \theta=\pi/2-\tan^{-1}(7/24),\\ 0\leq\theta\leq\pi/2.\)
