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3 years ago

Determine the measure of angle A, B, and C in triangle ABC. If m∠A=(x-10)°,m∠B=(2x-50)°,and m∠C=x°

Mathematics

1 answer:

Effectus [21]3 years ago

7 0

Answer:

m∠A = 50°

m∠B = 70°

m∠C = 60°

Step-by-step explanation:

Determine the measure of angle A, B, and C in triangle ABC. If m∠A=(x-10)°,m∠B=(2x-50)°,and m∠C=x°

In a Triangle, the sum of the interior angles of a triangle = 180°

Step 1

We solve for x

Hence:

m∠A + m∠B + m∠C= 180°

(x-10)°+ (2x-50)°+ x° = 180°

x - 10 + 2x - 50 + x = 180°

4x - 60 = 180°

4x = 180° + 60°

4x = 240°

x = 240°/4

x = 60°

Step 2

Solving for each measure

x = 60°

m∠A=(x-10)°

= 60° - 10°

= 50°

m∠B=(2x-50)°

= 2(60)° - 50°

= 120° - 50°

= 70°

m∠C=x°

= 60°

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4. A small high school holds its graduation ceremony in the gym. Because of seating constraints, students are limited to a maxim

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Answer:

(a) The mean and standard deviation of X is 2.6 and 1.2 respectively.

(b) The mean and standard deviation of T are 390 and 180 respectively.

(c) The distribution of T is N (390, 180²). The probability that all students’ requests can be accommodated is 0.7291.

Step-by-step explanation:

(a)

The random variable X is defined as the number of tickets requested by a randomly selected graduating student.

The probability distribution of the number of tickets wanted by the students for the graduation ceremony is as follows:

X P (X)

0 0.05

1 0.15

2 0.25

3 0.25

4 0.30

The formula to compute the mean is:

$\mu=\sum x\cdot P(X)$

Compute the mean number of tickets requested by a student as follows:

$\mu=\sum x\cdot P(X)\\=(0\times 0.05)+(1\times 0.15)+(2\times 0.25)+(3\times 0.25)+(4\times 0.30)\\=2.6$

The formula of standard deviation of the number of tickets requested by a student as follows:

$\sigma=\sqrt{E(X^{2})-\mu^{2}}$

Compute the standard deviation as follows:

$\sigma=\sqrt{E(X^{2})-\mu^{2}}\\=\sqrt{[(0^{2}\times 0.05)+(1^{2}\times 0.15)+(2^{2}\times 0.25)+(3^{2}\times 0.25)+(4^{2}\times 0.30)]-(2.6)^{2}}\\=\sqrt{1.44}\\=1.2$

Thus, the mean and standard deviation of X is 2.6 and 1.2 respectively.

(b)

The random variable T is defined as the total number of tickets requested by the 150 students graduating this year.

That is, T = 150 X

Compute the mean of T as follows:

$\mu=E(T)\\=E(150\cdot X)\\=150\times E(X)\\=150\times 2.6\\=390$

Compute the standard deviation of T as follows:

$\sigma=SD(T)\\=SD(150\cdot X)\\=\sqrt{V(150\cdot X)}\\=\sqrt{150^{2}}\times SD(X)\\=150\times 1.2\\=180$

Thus, the mean and standard deviation of T are 390 and 180 respectively.

(c)

The maximum number of seats at the gym is, 500.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sum of values of X, i.e ∑X, will be approximately normally distributed.

Here T = total number of seats requested.

Then, the mean of the distribution of the sum of values of X is given by,

$\mu_{T}=n\times \mu_{X}=390$