$MathMaster logo$

MathMaster

Home
general
a forest ranger sights a fire to the south a second ranger 8 miles east of the first ranger also sights the fire the beari

Question

A forest ranger sights a fire to the South. A second ranger, 8 miles East of the first ranger, also sights the fire. The bearing from the second ranger to the fire is S 33(degrees). How far is the first ranger from the fire?

67

likes

334 views

Answer to a math question A forest ranger sights a fire to the South. A second ranger, 8 miles East of the first ranger, also sights the fire. The bearing from the second ranger to the fire is S 33(degrees). How far is the first ranger from the fire?

$Expert avatar$

4.7

94 Answers

Let's denote the position of the first ranger as point A, the position of the fire as point F, and the position of the second ranger as point B.

Given:
AB = 8 miles
Angle FBA = 33 degrees

To find the distance between the first ranger (A) and the fire (F), denoted as AF, we can use trigonometry.

We first find the length of side BF using the cosine rule:
cos(33 degrees) = BF / AB
BF = AB * cos(33 degrees)
BF = 8 * cos(33 degrees)

Next, we can use the sine rule to find the distance AF:
sin(33 degrees) = AF / BF
AF = BF * sin(33 degrees)

Calculating the values:
BF = 8 * cos(33 degrees)
BF ≈ 8 * 0.8387
BF ≈ 6.71 miles

AF = BF * sin(33 degrees)
AF ≈ 6.71 * 0.5446
AF ≈ 3.65 miles

Therefore, the first ranger is approximately 3.65 miles away from the fire.

\boxed{AF \approx 3.65 \text{ miles}}

Frequently asked questions (FAQs)

What is the solution set of the inequality x + 5 > 10?

+

Math question: What is the value, in degrees, of the third angle in a triangle when the other two angles measure 45 degrees and 70 degrees?

+

Math question: Find the derivative of f(x) = 3x² + 4x - 5.

+

New questions in Mathematics

If we have the sequence: 3, 6, 12, 24 Please determine the 14th term.

Determine all solutions to the inequality |2x + 6| − |x + 1| < 6. Write your final answer in interval notation

A, B, C and D are numbers; If ABCD = 23, What is the result of ABCD BCDA CDAB DABC operation?

Suppose 50% of the doctors and hospital are surgeons if a sample of 576 doctors is selected what is the probability that the sample proportion of surgeons will be greater than 55% round your answer to four decimal places

What is the r.p.m. required to drill a 13/16" hole in mild steel if the cutting speed is 100 feet per minute?

7. Find the equation of the line passing through the points (−4,−2) 𝑎𝑛𝑑 (3,6), give the equation in the form 𝑎𝑥+𝑏𝑦+𝑐=0, where 𝑎,𝑏,𝑐 are whole numbers and 𝑎>0.

Suppose that you use 4.29 g of Iron in the chemical reaction: 2Fe(s) + 3 Cu2 + (aq) 2Fe 3 + (aq) + 3Cu(s ) - . What is the theoretical yield of Cu (s), in grams?

I. Order to add 40.25+1.31+.45 what is the first action to do ?

Find each coefficient described. Coefficient of u^2 in expansion of (u - 3)^3

In a laboratory test, it was found that a certain culture of bacteria develops in a favorable environment, doubling its population every 2 hours. The test started with a population of 100 bacteria. After six hours, it is estimated that the number of bacteria will be:

If a|-7 and a|9, then a|-63

X~N(2.6,1.44). find the P(X<3.1)

Twenty‐five students in a class take a test for which the average grade is 75. Then a twenty‐sixth student enters the class, takes the same test, and scores 70. The test average grade calculated with 26 students will

viii. An ac circuit with a 80 μF capacitor in series with a coil of resistance 16Ω and inductance 160mH is connected to a 100V, 100 Hz supply is shown below. Calculate 7. the inductive reactance 8. the capacitive reactance 9. the circuit impedance and V-I phase angle θ 10. the circuit current I 11. the phasor voltages VR, VL, VC and VS 12. the resonance circuit frequency Also construct a fully labeled and appropriately ‘scaled’ voltage phasor diagram.

The average weekly earnings in the leisure and hospitality industry group for a re‐ cent year was $273. A random sample of 40 workers showed weekly average ear‐ nings of $285 with the population standard deviation equal to 58. At the 0.05 level of significance can it be concluded that the mean differs from $273? Find a 95% con‐ fidence interval for the weekly earnings and show that it supports the results of the hypothesis test.

Today a father deposits $12,500 in a bank that pays 8% annual interest. Additionally, make annual contributions due of $2,000 annually for 3 years. The fund is for your son to receive an annuity and pay for his studies for 5 years. If the child starts college after 4 years, how much is the value of the annuity? solve how well it is for an exam

It costs a manufacturer $2,500 to purchase the tools to manufacture a certain homemade item. If the cost for materials and labor is 60¢ per item produced, and if the manufacturer can sell each item for 90¢, find how many items must he produce and sell to make a profit of $2000?

Exercise The temperature T in degrees Celsius of a chemical reaction is given as a function of time t, expressed in minutes, by the function defined on ¿ by: T (t )=(20 t +10)e−0.5t. 1) What is the initial temperature? 2) Show that T' (t )=(−10 t +15)e−0 .5t. 3) Study the sign of T' (t ), then draw up the table of variations of T . We do not ask for the limit of T in +∞. 4) What is the maximum temperature reached by the reaction chemical. We will give an approximate value to within 10−2. 5) After how long does the temperature T go back down to its initial value? We will give an approximate value of this time in minutes and seconds. DM 2: study of a function Exercise The temperature T in degrees Celsius of a chemical reaction is given as a function of time t, expressed in minutes, by the function defined on ¿ by: T (t )=(20 t +10)e−0.5t. 1) What is the initial temperature? 2) Show that T' (t )=(−10 t +15)e−0.5 t. 3) Study the sign of T' (t ), then draw up the table of variations of T . We do not ask for the limit of T in +∞. 4) What is the maximum temperature reached by the reaction chemical. We will give an approximate value to within 10−2. 5) After how long does the temperature T go back down to its initial value? We will give an approximate value of this time in minutes and seconds.