The bearing of P from Q is 072°. Find the bearing of Q from P. A number is selected at random from the set…

Question

The bearing of P from Q is 072°. Find the bearing of Q from P. A number is selected at random from the set M=2 x < 16. Find the probability that the number is a multiple of 5.

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6. A man 1.6 m tall stands 20 m from a building of height 27 m. Find, correct to one decimal place, the angle of elevation of the top of the building from the man.

Determine the height of the building above the man's eye level.
$\text{Effective height} = \text{Building height} - \text{Man's height}$ $\text{Effective height} = 27 \text{ m} - 1.6 \text{ m} = 25.4 \text{ m}$
Use trigonometry to find the angle of elevation.
The distance from the building is the adjacent side, and the effective height is the opposite side.
$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ $\tan \theta = \frac{25.4}{20}$ $\tan \theta = 1.27$
Calculate the angle $\theta$ .
$\theta = \arctan(1.27)$ $\theta \approx 51.78^\circ$
Rounding to one decimal place, $\theta \approx 51.8^\circ$ .

The correct option is A.
$\boxed{\text{A. } 51.8^\circ}$
7. In the diagram, $PR$ is tangent to the circle at $Q$ . $\angle SQT = 58^\circ$ and $\angle STQ = 69^\circ$ . Find $\angle PQR$ .
Find $\angle QST$ in $\triangle SQT$ .
The sum of angles in a triangle is $180^\circ$ .
$\angle QST + \angle SQT + \angle STQ = 180^\circ$ $\angle QST + 58^\circ + 69^\circ = 180^\circ$ $\angle QST + 127^\circ = 180^\circ$ $\angle QST = 180^\circ - 127^\circ = 53^\circ$
Apply the Alternate Segment Theorem.
The angle between a tangent ( $PR$ ) and a chord ( $SQ$ ) through the point of contact ( $Q$ ) is equal to the angle in the alternate segment ( $\angle QST$ ).
$\angle PQR = \angle QST$ $\angle PQR = 53^\circ$
The correct option is A.
$\boxed{\text{A. } 53^\circ}$
8. Given that $\tan R = \frac{7}{24}$ , where $0^\circ < R < 90^\circ$ , find $\sin R$ .
Use the given tangent value to find the hypotenuse of a right-angled triangle.
$\tan R = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{24}$
Let the opposite side be 7 and the adjacent side be 24.
Using the Pythagorean theorem, the hypotenuse $h$ is:
$h = \sqrt{\text{opposite}^2 + \text{adjacent}^2}$ $h = \sqrt{7^2 + 24^2}$ $h = \sqrt{49 + 576}$ $h = \sqrt{625}$ $h = 25$
Calculate $\sin R$ .
$\sin R = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin R = \frac{7}{25}$
The correct option is A.
$\boxed{\text{A. } \frac{7}{25}}$
9. The bearing of $P$ from $Q$ is $072^\circ$ . Find the bearing of $Q$ from $P$ .
Understand the relationship between forward and back bearings.
The bearing of $P$ from $Q$ is $072^\circ$ . This is the forward bearing.
Calculate the back bearing.
If the forward bearing is less than $180^\circ$ , add $180^\circ$ to find the back bearing.
$\text{Bearing of } Q \text{ from } P = \text{Bearing of } P \text{ from } Q + 180^\circ$ $\text{Bearing of } Q \text{ from } P = 072^\circ + 180^\circ$ $\text{Bearing of } Q \text{ from } P = 252^\circ$
The correct option is B.
$\boxed{\text{B. } 252^\circ}$
10. A number is selected at random from the set $M = \{x: 2 < x < 16\}$ . Find the probability that the number is a multiple of 5.
List the elements in set $M$ .
The set $M$ contains integers strictly greater than 2 and strictly less than 16.
$M = \{3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$
Count the total number of elements in $M$ .
$\text{Total number of elements} = 15 - 3 + 1 = 13$
Identify the multiples of 5 within set $M$ .
The multiples of 5 in $M$ are 5 and 10.
$\text{Number of multiples of 5} = 2$
Calculate the probability.
$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ $\text{Probability} = \frac{2}{13}$
The correct option is B.
$\boxed{\text{B. } \frac{2}{13}}$
11. Given that $y = 3x^2 - 7x + 10$ , find $\frac{dy}{dx}$ at $x=3$ .
Differentiate $y$ with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(3x^2) - \frac{d}{dx}(7x) + \frac{d}{dx}(10)$ $\frac{dy}{dx} = 3(2x^{2-1}) - 7(1x^{1-1}) + 0$ $\frac{dy}{dx} = 6x - 7$
Substitute $x=3$ into the derivative.
$\frac{dy}{dx}\Big|_{x=3} = 6(3) - 7$ $\frac{dy}{dx}\Big|_{x=3} = 18 - 7$ $\frac{dy}{dx}\Big|_{x=3} = 11$
The correct option is B.
$\boxed{\text{B. } 11}$
12. The interior angle of a regular polygon is $140^\circ$ . Calculate the sum of the interior angles of the polygon.
Find the number of sides ( $n$ ) of the polygon using the formula for an interior angle of a regular polygon.
$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$ $140^\circ = \frac{(n-2) \times 180^\circ}{n}$ $140n = 180(n-2)$ $140n = 180n - 360$ $360 = 180n - 140n$ $360 = 40n$ $n = \frac{360}{40}$ $n = 9$
The polygon has 9 sides.
Calculate the sum of the interior angles using the formula $(n-2) \times 180^\circ$ .
$\text{Sum of interior angles} = (9-2) \times 180^\circ$ $\text{Sum of interior angles} = 7 \times 180^\circ$ $\text{Sum of interior angles} = 1260^\circ$
The correct option is B.
$\boxed{\text{B. } 1260^\circ}$
13. Given that $\frac{a}{b} = \frac{1}{p} - \frac{1}{q}$ , make $p$ the subject.
(Assuming the question intended to ask to make $p$ the subject, as 'C' is not present in the equation and the options are for $p$ .)
Isolate the term containing $p$ .
$\frac{a}{b} = \frac{1}{p} - \frac{1}{q}$
Add $\frac{1}{q}$ to both sides:
$\frac{a}{b} + \frac{1}{q} = \frac{1}{p}$
Combine the terms on the left side by finding a common denominator, which is $bq$ .
$\frac{aq}{bq} + \frac{b}{bq} = \frac{1}{p}$ $\frac{aq + b}{bq} = \frac{1}{p}$
Invert both sides to solve for $p$ .
$p = \frac{bq}{aq + b}$
The correct option is B.
$\boxed{\text{B. } p = \frac{bq}{aq + b}}$

The bearing of P from Q is 072°. Find the bearing of Q from P. A number is selected at random from the set M=2 x < 16. Find the probability that the number is a multiple of 5.

Answer

Step-by-step solution

Determine the height of the building above the man's eye level.

Use trigonometry to find the angle of elevation.

Calculate the angle $\theta$ .

Find $\angle QST$ in $\triangle SQT$ .

Apply the Alternate Segment Theorem.

Use the given tangent value to find the hypotenuse of a right-angled triangle.

Calculate $\sin R$ .

Understand the relationship between forward and back bearings.

Calculate the back bearing.

List the elements in set $M$ .

Count the total number of elements in $M$ .

Identify the multiples of 5 within set $M$ .

Calculate the probability.

Differentiate $y$ with respect to $x$ .

Substitute $x=3$ into the derivative.

Find the number of sides ( $n$ ) of the polygon using the formula for an interior angle of a regular polygon.

Calculate the sum of the interior angles using the formula $(n-2) \times 180^\circ$ .

Isolate the term containing $p$ .

Combine the terms on the left side by finding a common denominator, which is $bq$ .

Invert both sides to solve for $p$ .

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The bearing of P from Q is 072°. Find the bearing of Q from P. A number is selected at random from the set M=2 x < 16. Find the probability that the number is a multiple of 5.

Answer

Step-by-step solution

Determine the height of the building above the man's eye level.

Use trigonometry to find the angle of elevation.

Calculate the angle θ\thetaθ.

Find ∠QST\angle QST∠QST in △SQT\triangle SQT△SQT.

Apply the Alternate Segment Theorem.

Use the given tangent value to find the hypotenuse of a right-angled triangle.

Calculate sin⁡R\sin RsinR.

Understand the relationship between forward and back bearings.

Calculate the back bearing.

List the elements in set MMM.

Count the total number of elements in MMM.

Identify the multiples of 5 within set MMM.

Calculate the probability.

Differentiate yyy with respect to xxx.

Substitute x=3x=3x=3 into the derivative.

Find the number of sides (nnn) of the polygon using the formula for an interior angle of a regular polygon.

Calculate the sum of the interior angles using the formula (n−2)×180∘(n-2) \times 180^\circ(n−2)×180∘.

Isolate the term containing ppp.

Combine the terms on the left side by finding a common denominator, which is bqbqbq.

Invert both sides to solve for ppp.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate the angle $\theta$ .

Find $\angle QST$ in $\triangle SQT$ .

Calculate $\sin R$ .

List the elements in set $M$ .

Count the total number of elements in $M$ .

Identify the multiples of 5 within set $M$ .

Differentiate $y$ with respect to $x$ .

Substitute $x=3$ into the derivative.

Find the number of sides ( $n$ ) of the polygon using the formula for an interior angle of a regular polygon.

Calculate the sum of the interior angles using the formula $(n-2) \times 180^\circ$ .

Isolate the term containing $p$ .

Combine the terms on the left side by finding a common denominator, which is $bq$ .

Invert both sides to solve for $p$ .