Hi FBSK

Please find below the solution to your first question:

Since this question talks about circumference and Surface Area of a sphere, let's first derive the relationship between the two.

Circumference of a Sphere, \(C = 2 pie R\), where R is the radius of the sphere.

So,\(R=\frac{C}{2 pie}\)

Surface Area of a Sphere, \(S = 4 pie R^2\)

Substituting for the value of R,

\(S= 4 pie\frac{C^2}{(2 pie)^2}\)

So, \(S= \frac{C^2}{pie}\)

This is the result that we will use to solve this question.

For Sphere 1,

C= 5.5 m

So, \(S = \frac{(5.5)^2}{pie}\)

Put \(pie =\frac{22}{7}\)

So,\(S = \frac{55*55*7}{10*10*22}\)

\(S = \frac{77}{8}\)sq. m

So, Cost of painting the surface area of Sphere 1 = \(92*\frac{77}{8}\)

=\(23*\frac{77}{2}\)

Now, here I want you to be smart. Look at the values in the five options. They are all quite far apart from each other. This means, that you don't need to find the exact value, true to the first or the second place of decimal. You only need a ballpark estimate of the cost.

So, to quickly do this rough estimate, let's replace 23 with 22. You can appreciate how easy our calculation now becomes!

\(11*77 = 847\)

The option closest to this is 900. So, our answer for the first sphere will be 900.

Now, coming to the second sphere:

We'll again use the formula \(S= \frac{C^2}{pie}\)

Here, \(S = \frac{(7.85)^2}{pie}\)

Seems like a complex calculation, huh?

Let's just hold on the calculation for a minute and put in the formula for costs.

Cost of painting Sphere 2 = \(92* \frac{(7.85)^2}{pie}\)

= \(92*7.85*7.85*\frac{7}{22}\)

Now we'll simplify these decimals to get an approximate value of the cost:

Cost = \(4*8*8*7\) (because \(\frac{92}{22}\) is approximately equal to 4)

= 1792

The closest option to this is 1800. So, we'll choose it as our answer.

Hope this helped!

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