Question Number 209956 by Jubr last updated on 26/Jul/24
![Find the maximum value of 7cosA + 24sinA + 32](https://www.tinkutara.com/question/Q209956.png)
Find the maximum value of
7cosA + 24sinA + 32
7cosA + 24sinA + 32
Answered by Frix last updated on 27/Jul/24
![7≤7cos A +24sin A +32≤57](https://www.tinkutara.com/question/Q209959.png)
$$\mathrm{7}\leqslant\mathrm{7cos}\:{A}\:+\mathrm{24sin}\:{A}\:+\mathrm{32}\leqslant\mathrm{57} \\ $$
Commented by Jubr last updated on 28/Jul/24
![Thanks sir](https://www.tinkutara.com/question/Q209995.png)
$${Thanks}\:{sir} \\ $$
Answered by mr W last updated on 27/Jul/24
![=(√(7^2 +24^2 )) sin (A+tan^(−1) (7/(24)))+32 maximum=(√(7^2 +24^2 ))+32=25+32=57 minimum=−(√(7^2 +24^2 ))+32=−25+32=7](https://www.tinkutara.com/question/Q209964.png)
$$=\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }\:\mathrm{sin}\:\left({A}+\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{7}}{\mathrm{24}}\right)+\mathrm{32} \\ $$$${maximum}=\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }+\mathrm{32}=\mathrm{25}+\mathrm{32}=\mathrm{57} \\ $$$${minimum}=−\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }+\mathrm{32}=−\mathrm{25}+\mathrm{32}=\mathrm{7} \\ $$
Commented by Jubr last updated on 28/Jul/24
![Thanks sir](https://www.tinkutara.com/question/Q209996.png)
$${Thanks}\:{sir} \\ $$
Answered by A5T last updated on 27/Jul/24
![7cosA+24sinA+32≤(√(7^2 +24^2 ))(√(sin^2 A+cos^2 A))+32 =25+32=57](https://www.tinkutara.com/question/Q209966.png)
$$\mathrm{7}{cosA}+\mathrm{24}{sinA}+\mathrm{32}\leqslant\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{24}^{\mathrm{2}} }\sqrt{{sin}^{\mathrm{2}} {A}+{cos}^{\mathrm{2}} {A}}+\mathrm{32} \\ $$$$=\mathrm{25}+\mathrm{32}=\mathrm{57} \\ $$
Commented by Jubr last updated on 28/Jul/24
![Thanks sir](https://www.tinkutara.com/question/Q209997.png)
$${Thanks}\:{sir} \\ $$