Question

A box contains an
even number of solid
spheres. half of
them are split into
equal hemispheres
and then, half of the
hemispheres are
divided into equal
identical halves, by
what percentage
would the cost of
painting increase as
compared
to
painting the spheres
without cutting?​

Answer 1

Answer:

A hemisphere is the half sphere formed by a plane intersecting the center of a sphere. The cut-line forming a hemisphere is a great circle. Volume of a Sphere: ... The volume inside of a sphere is four-thirds times π, times the cube of its radius.

Answer 2

Answer:

The percentage increase in the cost of painting = 25%

Step-by-step explanation:

Let the number of solid spheres = 2x (given it is even)

After cutting half of the spheres,

No of solid spheres = x

No. of hemispheres = 2x

To find the cost of painting, we need to find the total surface area.

We know,

The total surface area of a sphere = $4\pi r^{2}$

The total surface area of a hemisphere = $3\pi r^{2}$

Before Cutting

No of spheres  = 2x

Total surface area of '2x' spheres =  2x * $4\pi r^{2}$

= $8x \pi r^{2}$

After Cutting

No of spheres = x

No of hemispheres = 2x

Total surface area = Total surface area of 'x' spheres + Total surface area of '2x' hemishperes

= $x*4\pi r ^{2} + 2x * 3\pi r^{2}$

= $4x\pi r ^{2} + 6x\pi r^{2}$

= $10x\pi r ^{2}$

Hence we have,

Total surface area before cutting =  = $8x \pi r^{2}$

Total surface area after cutting    = $10x\pi r ^{2}$

Required to find the percentage increase in the cost of painting

We know the percentage increase

=  $\frac{ final value - initial value}{initial value}$*100%

Here, the percentage increase in the cost of painting

= $\frac{Total surface area after cutting - total surface are before cutting}{ total surface are before cutting}$ *100%

= $\frac{10x\pi r ^{2} - 8x\pi r ^{2} \\}{ 8x\pi r ^{2} \\}$ *100%

=    $\frac{2x\pi r ^{2} }{8x\pi r ^{2} }$ *100%

= $\frac{1}{4}$ * 100%

= 25%

Hence the percentage increase in the cost of painting = 25%