To determine the value of [tex]\(a\)[/tex], which represents the number of times meringue was ordered, given that the probability of ordering meringue is 0.312, we follow these steps:
1. Calculate the total number of orders for the other desserts:
[tex]\[
\text{Total orders excluding meringue} = \text{Panna cotta} + \text{Brownie} + \text{Chocolate soufflé}
\][/tex]
[tex]\[
= 36 + 28 + 22
\][/tex]
[tex]\[
= 86
\][/tex]
2. Let [tex]\( T \)[/tex] be the total number of orders including meringue. Therefore, the probability of ordering meringue can be expressed as:
[tex]\[
\frac{a}{T} = 0.312
\][/tex]
3. Express [tex]\( T \)[/tex] in terms of [tex]\( a \)[/tex] using the total orders:
[tex]\[
T = 86 + a
\][/tex]
4. Substitute [tex]\( T \)[/tex] into the probability equation:
[tex]\[
\frac{a}{86 + a} = 0.312
\][/tex]
5. Solve for [tex]\( a \)[/tex] by isolating [tex]\( a \)[/tex]:
[tex]\[
a = 0.312 \times (86 + a)
\][/tex]
[tex]\[
a = 26.832 + 0.312a
\][/tex]
[tex]\[
a - 0.312a = 26.832
\][/tex]
[tex]\[
0.688a = 26.832
\][/tex]
[tex]\[
a = \frac{26.832}{0.688}
\][/tex]
[tex]\[
a \approx 39
\][/tex]
6. Match the calculated value of [tex]\( a \)[/tex] with the given choices: The value closest to 39 is indeed one of the choices provided.
Therefore, the value of [tex]\(a\)[/tex] is [tex]\(39\)[/tex].
The correct answer is [tex]\( \boxed{39} \)[/tex].
