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Setler [38]
3 years ago
11

What is the domain and range of this function?

Mathematics
1 answer:
max2010maxim [7]3 years ago
5 0

As for the domain, the only restriction comes from the logarithm. The argument of a logarith must be strictly positive, so we have

x-2>0 \iff x>2

As for the range, we have:

  • The range of \log(x) are all real numbers
  • If we change to \log(x-2) we're translating the function horizontally, so the range remains the same
  • If we change to \log(5(x-2() we're stretching the function horizontally, so the range doesn't change
  • If we change to \log(5(x-2))+1 we're translating the function 1 unit up, but the range is already all the real numbers, so it doesn't change.

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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
3 years ago
Which fraction is equivalent
Alina [70]
The answer would be d
3 0
1 year ago
Read 2 more answers
1/4 (20x−28) <2(x−3)+2
xeze [42]

Answer:

Step-by-step explanation:

1/4 x 20x - 1/4 x 28<2x - 6 + 2

5x - 28/4<2x - 4

5x - 2x<-4 + 28/4

3x<-4/1 + 28/4

3x<-16 + 28/4

3x<12/4

3x<3

3x/3<3/3

x<1

6 0
3 years ago
A triangle is cut from a piece of fabric. The triangle has a height of 8 inches in an area of 120 in.². What is the length of th
andreyandreev [35.5K]

Answer: 30 inches

Step-by-step explanation:

Given:

ABC-tringle

BH= 8 in - height

S= 120 in^2

AC-?

Solution:

S=1/2ab

120in=1/2a*8in

120=4a

4a=120

a=30 in

8 0
3 years ago
Write 9020000 in standard form?
dolphi86 [110]

Answer:

9.02 x 10^6

Step-by-step explanation:

hope this helps!

8 0
3 years ago
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