Answer :
Answer:
The edge of a cube is decreasing at the rate of 0.04 cm/ sec. If the edge of the cube is 10 cm, then rate of decrease of surface area of the cube is 4.8 cm²/sec
Step-by-step explanation:
Let assume that edge of a cube be x cm.
Let further assume that S represents the surface of a cube.
Given that, edge of a square is decreasing at the rate of 0.04 cm/sec.
[tex]\implies\sf\:\dfrac{dx}{dt} =-\: 0.04\: cm \: per \: sec \\ [/tex]
We know, Surface area of a sphere (S) of edge x is given by
[tex]\sf\: S = 6{x}^{2} \\ [/tex]
On differentiating both sides w. r. t. t, we get
[tex]\sf\: \dfrac{d}{dt} S = \dfrac{d}{dt}6 {x}^{2} \\ [/tex]
[tex]\sf\: \dfrac{dS}{dt} = 12x\dfrac{dx}{dt} \\ [/tex]
On substituting the values, we get
[tex]\sf\: \dfrac{dS}{dt} = 12 \times 10 \times (-0.04)\\ [/tex]
[tex]\sf\: \dfrac{dS}{dt} = -\:120 \times \dfrac{4}{100} \\ [/tex]
[tex]\sf\: \dfrac{dS}{dt} = -\:12 \times \dfrac{4}{10} \\ [/tex]
[tex]\implies\sf\: \dfrac{dS}{dt} = -\:4.8 \: {cm}^{2} \: per \: sec \\ [/tex]
Hence, surface area of a cube is decreasing at the rate of 4.8 cm²/sec
[tex] \mathfrak{\huge{\red{\underline{\underline{Answer :}}}}}[/tex]
The edge of a cube is decreasing at the rate of 0.04 cm/ sec. If the edge of the cube is 10 cm, then rate of decrease of surface area of the cube is
[tex] \color{red}\bf \underline { 4.8 cm²/sec}[/tex]
[tex]\color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}}\color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}}\color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt} \huge { \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{4pt}{9pt}} \color{red}\rule{4pt}{9pt} \color{orange}\rule{4pt}{9pt} \color{yellow}\rule{4pt}{9pt} \color{lime}\rule{4pt}{9pt} \color{blue}\rule{4pt}{9pt} \color{orchid}\rule{4pt}{9pt} \color{pink}\rule{999pt}{9pt} [/tex]
[tex]{\huge{\colorbox {babypink}{ʂoluէìօղ}}}[/tex]
[tex]\bf\: S = 6{x}^{2} \\
\bf\: \dfrac{d}{dt} S = \dfrac{d}{dt}6 {x}^{2} \\ [/tex]
[tex]\bf\: \dfrac{dS}{dt} = 12x\dfrac{dx}{dt} \\ [/tex]
[tex]\bf\: \dfrac{dS}{dt} = 12 \times 10 \times (-0.04)\\ [/tex]
[tex]\bf\: \dfrac{dS}{dt} = -\:120 \times \dfrac{4}{100} \\ [/tex]
[tex]\bf\: \dfrac{dS}{dt} = -\:12 \times \dfrac{4}{10} \\ \\ [/tex]
[tex]\underline{\bf\: \dfrac{dS}{dt} = -\:4.8 \: {cm}^{2} \: per \: sec } \\
[/tex]
[tex] \color{red}\bf \underline { 4.8 cm²/sec}[/tex]