Find-the-maximum-value-of-7cosA-24sinA-32-

Question Number 209956 by Jubr last updated on 26/Jul/24

Find the maximum value of
7cosA + 24sinA + 32

Answered by Frix last updated on 27/Jul/24

$$\mathrm{7}\leqslant\mathrm{7cos}\:{A}\:+\mathrm{24sin}\:{A}\:+\mathrm{32}\leqslant\mathrm{57} \\ $$

Commented by Jubr last updated on 28/Jul/24

$${Thanks}\:{sir} \\ $$

Answered by mr W last updated on 27/Jul/24

$$=\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }\:\mathrm{sin}\:\left({A}+\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{7}}{\mathrm{24}}\right)+\mathrm{32} \\ $$$${maximum}=\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }+\mathrm{32}=\mathrm{25}+\mathrm{32}=\mathrm{57} \\ $$$${minimum}=−\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }+\mathrm{32}=−\mathrm{25}+\mathrm{32}=\mathrm{7} \\ $$

Commented by Jubr last updated on 28/Jul/24

$${Thanks}\:{sir} \\ $$

Answered by A5T last updated on 27/Jul/24

$$\mathrm{7}{cosA}+\mathrm{24}{sinA}+\mathrm{32}\leqslant\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }\sqrt{{sin}^{\mathrm{2}} {A}+{cos}^{\mathrm{2}} {A}}+\mathrm{32} \\ $$$$=\mathrm{25}+\mathrm{32}=\mathrm{57} \\ $$

Commented by Jubr last updated on 28/Jul/24

$${Thanks}\:{sir} \\ $$

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