The3Mathketeers

Since a, b, c, d, and e are all variations of a similar question, I think it is fair just to do problems a and d and leave the rest.

Expected value is the average outcome of running a long-term experiment, which is calculated with the formula \(E(X) = \sum_{i = 1}^n x_iP(x_i)\) where E(X) is the expected value, x_i is a specific outcome, and P(x_i) is the probability of x_i occurring. Let's apply this to problem a.

Upon drawing a slip of paper from the bag, 2 possible outcomes can occur: 3 or 8 with their respective probabilities. With the unmodified bag, 3 occurs with a probability of 8/10 because there are eight 3's and ten slips of paper. Likewise, 8 occurs with a probability of 2/10 because there are two 8's and ten slips of paper. Substitute this information into the formula and the answer follows naturally.

\(\begin{align} E(X) &= 3 * \frac{8}{10} + 8 * \frac{2}{10} \\ &= \frac{24}{10} + \frac{16}{10} \\ &= 4 \end{align}\)

Therefore, E(X) = 4 for problem a.

Next, let's consider problem d. Here, the expected value is given, so E(X) = 6. Adding 8's to the original contents of the bag increases both the number of 8's and the total number of 8's in the bag. Let a represent the number of 8's to add to the bag such that E(X) = 6. This would result in the following equation.

\(\begin{align*} E(X) &= \sum_{i = 1}^n x_iP(x_i) \\ 6 &= 3 * \frac{8}{10 + a} + 8 * \frac{2 + a}{10 + a} \\ 60 + 6a &= 24 + 16 + 8a \\ 20 &= 2a \\ a &= 10 \end{align*}\)

Therefore, add ten 8's to the bag. To be a proper mathematics steward, note that 10 + a appears in the denominator. This means a ≠ -10 under any circumstance. However, this is a non-issue regardless because the variable a represents the number of 8's added, so a negative value for a is nonsensical anyway.

The given quadratic equation is \(ax^2 + 8x + 4 = 6x - 40\). First, simplify it to standard form.

\(ax^2 + 8x + 4 = 6x - 40 \\ ax^2 + 2x + 44 = 0\)

Now, we can use the discriminant of a quadratic, which gives information about the number of solutions. In order for the quadratic to have only one solution, the discriminant should be equal to 0.

\(D = 0 \\ b^2 - 4ac = 0 \\ 2^2 - 4 * a * 44 \\ 4 - 176a = 0 \\ 176a = 4 \\ a = \frac{4}{176} \\ a = \frac{1}{44}\)

According to the given statement, \(\triangle {\text{PQR}} \cong \triangle {\text{STU}}\). Corresponding angles of congruent triangles have equal measure. Therefore, since \(m \angle \text{S} = 78^{\circ}\), this means that \(m \angle \text{P} = 78^{\circ}\). By the widely known Triangle Sum Angle Theorem, the sum of the interior angles of the triangle equals 180 degrees.

\(m \angle {\text{P}} + m \angle {\text{Q}} + m \angle {\text{R}} = 180^{\circ} \\ 78^{\circ} + m \angle {\text{Q}} + 80^{\circ} = 180^{\circ} \\ 158^{\circ} + m \angle {\text{Q}} = 180^{\circ} \\ m \angle {\text{Q}} = 22^{\circ}\)

Strictly speaking, an exponent tower is one of the few operations in mathematics that is evaluated from the "top" of the tower to the "bottom" of the tower. Unfortunately, this number is too large to write down. Although, we can attempt to write an approximation.

\({{3^3}^3}^3 = {3^3}^{27} = 3^{7\;625\;597\;484\;987} \approx {10^{10}}^{12.5609}\). The exponent is too large, so this number cannot be written down realistically. This number has about one trillion digits, which is far too many.

This question is a bit too broad, but I assume you want to simplify the expression \(5(3x^2 + 9x) - 14\) as much as possible? If so, there is not much to do to simplify this particular expression other than do the expansion within the parentheses. 

\(5(3x^2 + 9x) - 14 = 15x^2 + 45x - 14\)

Saying that the arithmetic mean of x and y is 18 mathematically means that \(\frac{x + y}{2} = 18\). Also, saying that x and y have a geometric mean of \(\sqrt{47}\) means that \(\sqrt{xy} = \sqrt{47}\). With this information, we want to find the value of \(x^2 + y^2\). We can use clever algebraic manipulation to find this value without too much trouble.

\(\frac{x + y}{2} = 18; \sqrt{xy} = \sqrt{47} \\ x + y = 36; xy = 47 \\ (x + y)^2 = 36^2 \\ x^2 + 2xy + y^2 = 1296 \\ x^2 + y^2 = 1296 - 2xy \\ x^2 + y^2 = 1296 - 2 * 47 \\ x^2 + y^2 = 1202 \)

Sometimes, it is hard to wrap one's head around how to determine how one quantity is related to another without a visual picture. As a result, I decided to create a Venn diagram that should make the situation clearer. I have included all the given information in the Venn diagram.

The given information states that there are twice as many students in the cooking club as in the drama club. Since we are using the a-variable to represent the number of students in the drama club, this means that 2a students are in the cooking club. We are letting the b-variable be in the middle; this represents the students who are members of both clubs. Our objective is to find an expression that represents the students in the one of the two clubs but not both.

I will start with the cooking club. The cooking club has 2a members. With the help of the diagram, it is clear there are b students who are members of both. Therefore, 2a - b represents the number of students who are only apart of the cooking club.

Considering the drama club, there are a total members. There are b students who are members of both. Therefore, a - b represents the number of students only participating in the drama club.

Now, we just find the sum of these parts. \(2a - b + a - b = 3a - 2b\) students who are members of one of the clubs but not both.