A circular cake is divided into slices where the angle of each slice grows by 22% compared to the previous slice. The smallest slice has an angle of 10°. The total sum of all slice... A circular cake is divided into slices where the angle of each slice grows by 22% compared to the previous slice. The smallest slice has an angle of 10°. The total sum of all slice angles is 360°. The cake has a diameter of 10 inches. Every 4th slice is taken immediately after cutting. Compute: The total number of slices (rounded to the nearest slice). The remaining cake area after all slices are cut and every 4th slice is taken (round to nearest whole number). Read more

Steps to Solve

1. Calculate the total number of slices

The total angle of a circle is $360^\circ$. We can represent the angles of each slice as follows: the first slice has an angle of $a_1$, the second slice has an angle of $a_2 = a_1 + d$, the third slice has $a_3 = a_1 + 2d$, and so on, where $d$ is the difference in angle between successive slices.

Assuming you have the first angle $a_1$ and the difference $d$, the angle for the nth slice can be represented by

$$ a_n = a_1 + (n-1)d $$

We need to find $n$ such that the total angle of all slices does not exceed $360^\circ$.

2. Sum the angles of the slices

The sum of the angles for the first n slices can be calculated using the formula for the sum of an arithmetic series:

$$ S_n = \frac{n}{2} (a_1 + a_n) $$

Replacing $a_n$ in the equation gives:

$$ S_n = \frac{n}{2} \left(a_1 + (a_1 + (n-1)d)\right) = \frac{n}{2}(2a_1 + (n-1)d) $$

We need to solve for n such that

$$ S_n \leq 360 $$

3. Identify the slices to be removed

Since we are removing every 4th slice, we need to calculate how many slices we are left with after that. If we have calculated $n$, then the number of removed slices is:

$$ \text{Removed Slices} = \frac{n}{4} $$

4. Calculate the area of the cake

The area of the cake can be calculated using the formula for the area of a circle:

$$ A = \pi r^2 $$

where $r$ is the radius. If the diameter $d$ is provided, then $r = \frac{d}{2}$.

5. Calculate the area removed

To calculate the total area removed (for every 4th slice), we need to find the angle of each slice being removed and convert that angle to area proportionate to the whole cake area.

The area of one slice is given by the formula:

$$ A_{\text{slice}} = \frac{\theta}{360} \times A $$

where $\theta$ is the angle of the slice.

The total area removed is given by:

$$ A_{\text{removed}} = \text{Removed Slices} \times A_{\text{slice}} $$

6. Calculate the remaining area

Finally, the remaining area of the cake is:

$$ A_{\text{remaining}} = A - A_{\text{removed}} $$