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the volume of two similar solids are 1080cm and 1715cm .if the curved surface area of the smaller cone is 840cm .fond the curved surface area of the larger cone​

Sagot :

MrRoyalidn

Answer:

[tex]A_{big} = 1143.33cm^2[/tex]

Explanation:

The given parameters are:

[tex]V_{small} = 1080[/tex]

[tex]V_{big} = 1715[/tex]

[tex]C_{small} = 840[/tex]

Required

Determine the curved surface area of the big cone

The volume of a cone is:

[tex]V = \frac{1}{3}\pi r^2h[/tex]

For the big cone:

[tex]V_{big} = \frac{1}{3}\pi R^2H[/tex]

Where

R = radius of the big cone and H = height of the big cone

For the small cone:

[tex]V_{small} = \frac{1}{3}\pi r^2h[/tex]

Where

r = radius of the small cone and H = height of the small cone

Because both cones are similar, then:

[tex]\frac{H}{h} = \frac{R}{r}[/tex]

and

[tex]\frac{V_{big}}{V_{small}} = \frac{\frac{1}{3}\pi R^2H}{\frac{1}{3}\pi r^2h}[/tex]

[tex]\frac{V_{big}}{V_{small}} = \frac{R^2H}{r^2h}[/tex]

Substitute values for Vbig and  Vsmall

[tex]\frac{1715}{1080} = \frac{R^2H}{r^2h}[/tex]

Recall that:[tex]\frac{H}{h} = \frac{R}{r}[/tex]

So, we have:

[tex]\frac{1715}{1080} = \frac{R^2*R}{r^2*r}[/tex]

[tex]\frac{1715}{1080} = \frac{R^3}{r^3}[/tex]

Take cube roots of both sides

[tex]\sqrt[3]{\frac{1715}{1080}} = \frac{R}{r}[/tex]

Factorize

[tex]\sqrt[3]{\frac{343*5}{216*5}} = \frac{R}{r}[/tex]

[tex]\sqrt[3]{\frac{343}{216}} = \frac{R}{r}[/tex]

[tex]\frac{7}{6} = \frac{R}{r}[/tex]

The curved surface area is calculated as:

[tex]Area = \pi rl[/tex]

Where

[tex]l = slant\ height[/tex]

For the big cone:

[tex]A_{big} = \pi RL[/tex]

For the small cone

[tex]A_{small} = \pi rl[/tex]

Because both cones are similar, then:

[tex]\frac{L}{l} = \frac{R}{r}[/tex]

and

[tex]\frac{A_{big}}{A_{small}} = \frac{\pi RL}{\pi rl}[/tex]

[tex]\frac{A_{big}}{A_{small}} = \frac{RL}{rl}[/tex]

This gives:

[tex]\frac{A_{big}}{A_{small}} = \frac{R}{r} * \frac{L}{l}[/tex]

Recall that:

[tex]\frac{L}{l} = \frac{R}{r}[/tex]

So, we have:

[tex]\frac{A_{big}}{A_{small}} = \frac{R}{r} * \frac{R}{r}[/tex]

[tex]\frac{A_{big}}{A_{small}} = (\frac{R}{r})^2[/tex]

Make [tex]A_{big}[/tex] the subject

[tex]A_{big} = (\frac{R}{r})^2 * A_{small}[/tex]

Substitute values for [tex]\frac{R}{r}[/tex] and [tex]A_{small}[/tex]

[tex]A_{big} = (\frac{7}{6})^2 * 840[/tex]

[tex]A_{big} = \frac{49}{36} * 840[/tex]

[tex]A_{big} = \frac{49* 840}{36}[/tex]

[tex]A_{big} = 1143.33cm^2[/tex]

Hence, the curved surface area of the big cone is 1143.33cm^2

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