Evaluate 6ᵗʷᵒ – (9 ÷ x) when x = 3.

Two lighthouses are located 75 miles from one another on a north-south line. If a boat is spotted S 40o E from the northern ligh

yuradex [85]

Answer:

The northern lighthouse is approximately $24.4\; \rm mi$ closer to the boat than the southern lighthouse.

Step-by-step explanation:

Refer to the diagram attached. Denote the northern lighthouse as $\rm N$ , the southern lighthouse as $\rm S$ , and the boat as $\rm B$ . These three points would form a triangle.

It is given that two of the angles of this triangle measure $40^{\circ}$ (northern lighthouse, $\angle {\rm N}$ ) and $21^{\circ}$ (southern lighthouse $\angle {\rm S}$ ), respectively. The three angles of any triangle add up to $180^{\circ}$ . Therefore, the third angle of this triangle would measure $180^{\circ} - (40^{\circ} + 21^{\circ}) = 119^{\circ}$ (boat $\angle {\rm B}$ .)

It is also given that the length between the two lighthouses (length of $\rm NS$ ) is $75\; \rm mi$ .

By the law of sine, the length of a side in a given triangle would be proportional to the angle opposite to that side. For example, in the triangle in this question, $\angle {\rm B}$ is opposite to side $\rm NS$ , whereas $\angle {\rm S}$ is opposite to side ${\rm NB}$ . Therefore:

$\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of NB}}{\sin(\angle {\rm S})} \end{aligned}$ .

Substitute in the known measurements:

$\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of NB}}{\sin(21^{\circ})} \end{aligned}$ .

Rearrange and solve for the length of $\rm NB$ :

$\begin{aligned} & \text{length of NB} \\ =\; & (75\; \rm mi) \times \frac{\sin(21^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 30.73\; \rm mi\end{aligned}$ .

(Round to at least one more decimal places than the values in the choices.)

Likewise, with $\angle {\rm N}$ is opposite to side ${\rm SB}$ , the following would also hold:

$\begin{aligned} \frac{\text{length of NS}}{\sin(\angle {\rm B})} = \frac{\text{length of SB}}{\sin(\angle {\rm N})} \end{aligned}$ .

$\begin{aligned} \frac{75\; \rm mi}{\sin(119^{\circ})} = \frac{\text{length of SB}}{\sin(40^{\circ})} \end{aligned}$ .

$\begin{aligned} & \text{length of SB} \\ =\; & (75\; \rm mi) \times \frac{\sin(40^{\circ})}{\sin(119^{\circ})} \\ \approx\; & 55.12\; \rm mi\end{aligned}$ .

In other words, the distance between the northern lighthouse and the boat is approximately $30.73\; \rm mi$ , whereas the distance between the southern lighthouse and the boat is approximately $55.12\; \rm mi$ . Hence the conclusion.

4 0

2 years ago

Using the properties of exponents and logarithms, find the value of x in 19 + 2 ln x = 25.

REY [17]

$19+2\ln x=25\ \ \ |-19\\\\2\ln x=6\ \ \ |:2\\\\\ln x=3\iff x=e^3$

$\text{Used de.finition of the logarithm:}\ \ \ \log_ab=c\iff a^c=b$

8 0

3 years ago

Read 2 more answers

Identify the coordinates of the point shown. An image of a coordinate plane with one point plotted. It is 3 units to the left of

Alex17521 [72]

Answer:

(-3, -6)

Step-by-step explanation:

always start from the origin! Hope this helps!

7 0

3 years ago

when a spherical balloon filled with air diameter of 6 inches what's the volume of balloon in cubic inches

Cerrena [4.2K]

Volume of sphere: V(s) = 4/3*pi*R^3 = (4/3)*pi*(D/2)^3 = (1/6) * pi * D^3

Volume of cube: V(c) = s^3

Volume of them is the same, I'm assuming you actually want to know the length of the cubic vertice

So s^3 = (1/6)*pi*D^3 -> s = (1/6 * pi)^1/3 * 6 = (36pi)^1/3

3 0

3 years ago

Read 2 more answers

X= ?? geometry ??? And how ????

steposvetlana [31]

Using the equations for tangent and secant lines:

Let the unknown length of the line inside the circle = y

6^2 = 3 x (y+3)

Simplify:

36 = 3y+9

Subtract 9 from both sides:

27 =3y

Divide both sides by 3:

Y = 9

Now add to get x:

X = 9 + 3 = 12

X = 12

8 0

3 years ago

Evaluate 6ᵗʷᵒ – (9 ÷ x) when x = 3.​

Evaluate 6ᵗʷᵒ – (9 ÷ x) when x = 3.