Chapter 12, Problem 18PT

To determine

To find the radius of new sphere dough.

Answer to Problem 18PT

5 centimeters (approximately)

Explanation of Solution

Given:

Two spherical cookie dough:

Radius ( $r_1$ ) $=3$ centimeters

Radius $(r_2)$$=5$ centimeters

Formula used:

Volume of a sphere $=\frac{4}{3}\pi r^3$

where r is the radius of the sphere.

Calculation:

Volume of sphere 1:

Let volume of sphere 1 be $V\text{'}$ .

  $\Rightarrow V\text{'}=\frac{4}{3}\pi r_1{}^3$

Putting the value of $r_1$ ,

  $\Rightarrow \frac{4}{3}\times 3.14\times 3\times 3\times 3$

  $\Rightarrow 37$ cubic centimeters

So, volume of sphere 1 is 37 cubic centimeters.

Volume of sphere 2:

Let volume of sphere 1 be $V\text{'}\text{'}$ .

  $\Rightarrow V\text{'}\text{'}=\frac{4}{3}\pi r_2{}^3$

Putting the value of $r_1$ ,

  $\Rightarrow \frac{4}{3}\times 3.14\times 5\times 5\times 5$

  $\Rightarrow 523$ cubic centimeters

So, volume of sphere 2 is 523 cubic centimeters.

Let volume of the bigger new sphere be $V$ .

Volume of big sphere $=$ Volume of sphere 1 $+$ Volume of sphere 2

  $\Rightarrow 37+523$

  $\Rightarrow 560$ cubic centimeters

Now,

Let $R$ be the radius of the big sphere.

  $V=\frac{4}{3}\pi R^3$

Putting the value of $V$ ,

  $\Rightarrow \frac{4}{3}\pi R^3=560$

  $\Rightarrow \frac{4}{3}\times 3.14\times R^3=560$

  $\Rightarrow R^3=\frac{560\times 3}{3.14\times 4}$

  $\Rightarrow R^3=133$

Taking cube root on both sides,

  $\Rightarrow R\approx 5$ centimeters

Conclusion:

Therefore, the radius of the big sphere is 5 centimeters (approximately).