Question

If the difference between the circumference and diameter of a circle is 30 cm, then the radius of the circle must be:
(A). 6 cm
(B). 7 cm
(C). 5 cm
(D). 8 cm

Answer 1

Hint: Assume the radius of the given circle as r. Now take the relation between radius, circumference and radius, diameter. Substitute these relations into the given relation between circumference and diameter. Now you have a single relation, solve this to get the value of r.

Complete step by step answer:
Relation between circumference and diameter, is as:
Circumference \[=diameter+30\] ……………………. (1)
We know the relations between the circumference, radius, and diameter:
Circumference $=2\pi \times \text{radius}$ …………………. (2)
Diameter $=2\left( \text{radius} \right)$ ……………… (3)
By assuming radius as variable and then substituting the equation (2), equation (3) in equation (1), we get it as –
$2\pi r =2r+30$
By subtracting the term 2r on both sides, we get –
$2\pi r-2r=2r-2r+30$
By simplifying the above equation, we get it as:
$2\pi r-2r=30$
By substituting value of $\pi $ as \[\dfrac{22}{7}\] , we get the equation as:
$2\dfrac{22}{7}r-2r=30$
By taking least common multiple on left hand side, we get:
$\dfrac{44r-14r}{7}=30$
By simplifying the above equation, we get it as:
$\dfrac{30r}{7}=30$
By multiplying by 7 on both sides of equations, we get:
$\dfrac{30r}{7}\times 7=30\times 7$
By simplifying the above equation, we get it as:
$30r=210$
By dividing with 30 on both sides of equation, we get:
$\dfrac{30r}{30}=\dfrac{210}{30}$
By simplifying the above equation, we get it as:
$r=7$
Therefore, the radius of the circle is 7cm.
Option (b) is correct.

Note: Be careful you must take 30 on the diameter side, because circumference has a factor of $\pi $ . By which we get that circumference is more than diameter. Alternate way is try to get all the ‘r’ terms to the right hand side and then solve, though you get the same result.