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Two sphere shaped products are manufactured. Their diameters are in the ratio of 1:2. What is the ratio of
surface areas needs paint?
✓ Marks: 1
x Negative Marks: 0.25​

Answer 1

Given:

Two products which are in the shape of the sphere are manufactured

The ratio of the their diameters are 1:2

To find:

The ratio of  their surface areas that needs to be painted

Formula to be Used:

$\boxed{Surface\: Area\: of\: the\: Sphere\: =\:4\:pi\:r^2}$

Solution:

Let the diameters of the two sphere shaped products be "d1" & "d2" and their radius be "r1" & "r2" respectively.

Since the ratio of the diameters is 1:2, so we can say, d1 = x and d2 = 2x.

Therefore,

Radius, r1 = $\frac{d1}{2}$ = $\frac{x}{2}$

and

Radius, r2 = $\frac{d2}{2}$ = $\frac{2x}{2}$ = $x$

Now,

Using the formula we will find the ratio of the surface area of the two spheres that needs to be painted.

∴ The ratio of the surface areas of the sphere needs to be painted is,

= [4πr1²] / [4πr2²]

we will substitute the values of r1 = $\frac{x}{2}$ and r2 = $x$

= [4 π $(\frac{x}{2} )^2$] / [4 π $x^2$]

we will now cancel the similar terms

= $\frac{\frac{1}{4}}{1}$

= $\bold{\frac{1}{4}}$

Thus, the ratio of surface areas of the spheres needs to be painted is 1:4.

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Answer 2

The ratio of  surface areas needs paint is 1 : 4

Step-by-step explanation:

The ratio of the diameter of the two spheres is 1 : 2

The surface area of the sphere is given by the formula:

A = 4π (d/2)²

The ratio of the surface area of two spheres is given as:

4π (d₁/2)² : 4π (d₂/2)²

The diameter of one sphere is x = d₁

The diameter of another sphere is 2x = d₂

Now, the ratio of surface area becomes,

A₁ : A₂ = 4π (x/2)² : 4π (2x/2)²

A₁ : A₂ = (x/2)² : (2x/2)²

A₁ : A₂ = x²/4 : 4x²/4

A₁ : A₂ = 4x² : 16x²

A₁ : A₂ = 4 : 16

∴ A₁ : A₂ = 1 : 4