Understanding Paper Roll Length Increase
When a long ream of paper with thickness $t$ is rolled tightly, each turn adds a layer of thickness $t$ to the radius of the roll.
Calculating Circumference Change
Let the radius of the roll be $r$ before adding a new turn. The length of the paper in this turn is approximately equal to the circumference, $C_1 = 2\pi r$.
After adding one full turn of paper (thickness $t$), the new radius becomes $r + t$. The length of the paper in this new, outer turn is approximately the new circumference, $C_2 = 2\pi (r + t)$.
Determining the Difference in Length
The difference in length between the current turn and the previous turn is:
$ \Delta L = C_2 - C_1 $
$ \Delta L = 2\pi (r + t) - 2\pi r $
$ \Delta L = 2\pi r + 2\pi t - 2\pi r $
$ \Delta L = 2\pi t $
Therefore, the length of the paper wrapped in one turn exceeds the length in the previous turn by $2\pi t$.
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- A pilgrim starts walking for a journey of 115 km. On the first day he covers 7 km, the next day 9 km, and likewise keeps adding 2 km everyday till he reaches 15 km per day which he maintains for the rest of the journey. How many days in all will he take to complete the journey?
If a, b and c are in Geometric Progression and $a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$ then, x, y, z are in ________.
Which of the following statement is true about the geometric series
$ 1 + r +r^2 + r^3 + ...............; (r > 0) $?
$6240$ रुपये की राशि $30$ किस्तों में इस प्रकार चुकाई जाती है कि प्रत्येक किस्त पिछली किस्त से $10$ रुपये अधिक है । पहली किस्त की मूल्य ____________है।
