1. Derive the half-angle formulas from the double
angle formulas.
2. Provide the formulas to convert between polar and
rectangular forms.
3. Convert one point from rectangular to polar and
another point from polar to rectangular.
4. Convert a rectangular equation to polar (rectangular
equation must contain squared x and y variables as
well as x and y variables raised to a single power)
and a polar equation to rectangular (polar equation
must contain an rand a ¦ (theta)).
Answers
1) cos (θ / 2) = √[(1 + cos θ) / 2], sin (θ / 2) = √[(1 - cos θ) / 2], tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) (x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°). The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) The linear function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
How to apply trigonometry on deriving formulas and transforming points
1) The following trigonometric formulae are used to derive the half-angle formulas:
sin² θ / 2 + cos² θ / 2 = 1 (1)
cos θ = cos² (θ / 2) - sin² (θ / 2) (2)
First, we derive the formula for the sine of a half angle:
cos θ = 2 · cos² (θ / 2) - 1
cos² (θ / 2) = (1 + cos θ) / 2
cos (θ / 2) = √[(1 + cos θ) / 2]
Second, we derive the formula for the cosine of a half angle:
cos θ = 1 - 2 · sin² (θ / 2)
2 · sin² (θ / 2) = 1 - cos θ
sin² (θ / 2) = (1 - cos θ) / 2
sin (θ / 2) = √[(1 - cos θ) / 2]
Third, we derive the formula for the tangent of a half angle:
tan (θ / 2) = sin (θ / 2) / cos (θ / 2)
tan (θ / 2) = √[(1 - cos θ) / (1 + cos θ)]
2) The formulae for the conversion of coordinates in rectangular form to polar form are obtained by trigonometric functions:
(x, y) → (r · cos θ, r · sin θ), where r = √(x² + y²).
3) Let be the point (x, y) = (2, 3), the coordinates in polar form are:
r = √(2² + 3²)
r = √13
θ = atan(3 / 2)
θ ≈ 56.309°
The point (x, y) = (2, 3) is equivalent to the point (r, θ) = (√13, 56.309°).
Let be the point (r, θ) = (4, 30°), the coordinates in rectangular form are:
(x, y) = (4 · cos 30°, 4 · sin 30°)
(x, y) = (2√3, 2)
The point (r, θ) = (4, 30°) is equivalent to the point (x, y) = (2√3, 2).
4) Let be the linear function y = 5 · x - 8, we proceed to use the following substitution formulas: x = r · cos θ, y = r · sin θ
r · sin θ = 5 · r · cos θ - 8
r · sin θ - 5 · r · cos θ = - 8
r · (sin θ - 5 · cos θ) = - 8
r = - 8 / (sin θ - 5 · cos θ)
The linear function y = 5 · x - 8 is equivalent to the function r = - 8 / (sin θ - 5 · cos θ).
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Related Questions
11. A number divided by -3 is equal to -2.8.
Answers
Solve for N (N =number ^ in your question) by simplifying both sides of the equation, then isolating the variable.
Gaurav was conducting a test to determine if the average amount of medication his patients were taking was similar to the national average. He wants to use a 5% significance level for his test to help ensure that his patients do not receive too little or too much medication. If Gaurav were to conduct a test, what probability value would indicate that his null hypothesis (that there is no significant difference between the amount of medication Gaurav's patients are receiving and the national average) would be rejected?
Answers
A probability value equal to or smaller than 0.05 would indicate that Gaurav's null hypothesis should be rejected at the 5% significance level.
In hypothesis testing, the significance level, denoted as alpha (α), is the predetermined threshold used to determine whether to reject the null hypothesis.
Gaurav has specified a 5% significance level, which means he wants to control the probability of making a Type I error (rejecting the null hypothesis when it is true) at 5% or less.
If Gaurav were to conduct a test and calculate the p-value, he would compare it to the significance level of 0.05.
The p-value is the probability of observing a test statistic as extreme as the one obtained, assuming the null hypothesis is true.
If the p-value is less than or equal to the significance level (p ≤ α), it indicates that the observed difference is unlikely to occur by chance alone under the assumption of the null hypothesis.
Gaurav would reject the null hypothesis and conclude that there is a significant difference between the average amount of medication his patients are taking and the national average.
Conversely, if the p-value is greater than the significance level (p > α), it suggests that the observed difference could reasonably occur by chance, and Gaurav would fail to reject the null hypothesis.
This would imply that there is no significant difference between the average medication amounts of Gaurav's patients and the national average.
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5) Express as a single logarithm. You do NOT need to expand the polynomials:
5 In(x - 2) + 8 In(x + 3) - 3 ln(x + 8)
Answers
5 ln(x - 2) + 8 ln(x + 3) - 3 ln(x + 8)
ln(x - 2)⁵ + ln(x + 3)⁸ - ln(x + 8)³
since \(a\log_cb=\log_cb^a\) (for any base c)
ln((x - 2)⁵ (x + 3)⁸ / (x + 8)³)
since \(\log_c(a)+\log_c(b)=\log_c(ab)\) and \(\log_c(a)-\log_c(b)=\log_c\frac ab\) (also for any base c)
BOWLING The cost for Nobu to go bowling is $4 per game plus an additional flat fee of $3.50 for the rental of bowling shoes. The cost can be modeled by the function f(x)=4x+3.5, where x represents the number of games bowled. Describe the graph of g(x) as it relates to f(x) if Nobu does not rent bowling shoes.
g(x) = , 1 of 3.
Select Choice
, which is the translation of f(x) , 2 of 3.
Select Choice
units , 3 of 3.
Select Choice
Answers
The graph of g(x) = 4x, is the translation of f(x) by subtracting 3.5 units, 2 of 3.
What is a graph?A graph can be defined as a pictorial representation or a diagram that represents data or values.
The cost can be modeled by the function f(x)=4x+3.5, where x represents the number of games bowled.
As per the given question, the required solution would be as:
g(x) = 4x , is the translation of f(x) by subtracting 3.5 units, 2 of 3.
The graph of g(x) would be identical to the graph of f(x), but shifted 3.5 units downward, 3 of 3.
This is because the flat fee for the rental of bowling shoes is not present in the function g(x), so the y-intercept of the graph would be at (0,-3.5) instead of (0,0) in f(x) function.
The slope and the shape of the function will remain the same.
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Write an equation of the line passing through the point (9,3) that is perpendicular to the line y = + 11.
Answers
Answer:
9=3+11
Step-by-step explanation:
The workers' union at a particular university is quite strong. About 96 of all workers employed by the university belong to the workers' union. Recently, the workers went on strike, and now a local TV station plans to interview 4 workers (chosen at random) at the university to get their opinions on the strike. What is the probability that exactly 2 of the workers interviewed are union members?
Answers
The probability that exactly 2 of the workers interviewed are union members is approximately 0.044.
To solve this problem, we can use the binomial probability formula, which is:
\(P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)\)
where:
n is the sample size (number of workers being interviewed)
k is the number of successes we are interested in (in this case, 2 of the workers being union members)
p is the probability of success (in this case, the probability that a worker is a union member)
(n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items (this can be calculated using the formula n! / (k! * (n - k)!))
Plugging in the values, we get:
\(P(X = 2) = (4 choose 2) * (0.96)^2 * (1 - 0.96)^(4 - 2)\)
\(= (6) * (0.96)^2 * (0.04)^2\)
= 0.04403136
Therefore, the probability that exactly 2 of the workers interviewed are union members is approximately 0.044.
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Which value below is included in the solution set for the inequality statement? -3(x-4) > 6(x-1) 0-1 02 07 0 3 NEXT QUESTION ASK FOR HELP
Answers
The solution set for the inequality is x < 2. Among the given options, the value that is included in the solution set is 0.
To determine which value is included in the solution set for the inequality statement -3(x-4) > 6(x-1), we need to solve the inequality for x.
Starting with the given inequality:
-3(x - 4) > 6(x - 1)
First, distribute -3 and 6 to the terms inside the parentheses:
-3x + 12 > 6x - 6
Next, combine like terms by subtracting 6x from both sides and adding 6 to both sides:
-3x - 6x > -6 - 12
-9x > -18
To isolate x, divide both sides of the inequality by -9. Remember that when dividing by a negative number, we need to reverse the inequality sign:
x < (-18) / (-9)
x < 2
Therefore, the solution set for the inequality is x < 2. Among the given options, the value that is included in the solution set is 0.
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The length of a rectangle is 5 cm more than its width. If the perimeter is 58cm, calculate:
(a) Write an equation to show the perimeter of the rectangle ?
(b) calculate:
I.width
II.length
III. the area of the rectangle
Answers
The equation to show the perimeter of the rectangle is P = 2(2w + 5)
Writing an equation to show the perimeter of the rectangleFrom the question, we have the following parameters that can be used in our computation:
Length = 5 more than the width
Also, we have
Perimeter = 58
This means that
P = 2(w + 5 + w)
P = 2(2w + 5)
Calculating the dimensions and the areaIn (a), we have
P = 2(2w + 5)
This gives
2(2w + 5) = 58
So, we have
2w + 5 = 29
2w = 24
w = 12
Next, we have
l = 12 + 5
l = 17
Lastly, we have
Area = 17 * 12
Area = 204
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Asap help please thanks!!!
Answers
Step-by-step explanation:
the exponent -1 (or any negative exponent) means 1/...
1/... of a fraction means to turn the fraction upside-down.
so
((5/7)² × (1/3)^-3)^-1 = (5/7)^-2 × (1/3)³ =
= (7/5)² × (1/3)³
The weights of newborn baby boys born at a local hospital are believed to have a normal distribution with a mean weight of 3950 grams and a standard deviation of 374 grams. If a newborn baby boy born at the local hospital is randomly selected, find the probability that the weight will be less than 4473 grams. Round your answer to four decimal places.
Answers
Answer:
0.9192 = 91.92% probability that the weight will be less than 4473 grams.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean \(\mu\) and standard deviation \(\sigma\), the z-score of a measure X is given by:
\(Z = \frac{X - \mu}{\sigma}\)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean weight of 3950 grams and a standard deviation of 374 grams.
This means that \(\mu = 3950, \sigma = 374\)
Find the probability that the weight will be less than 4473 grams.
This is the pvalue of Z when X = 4473. So
\(Z = \frac{X - \mu}{\sigma}\)
\(Z = \frac{4473 - 3950}{374}\)
\(Z = 1.4\)
\(Z = 1.4\) has a pvalue of 0.9192
0.9192 = 91.92% probability that the weight will be less than 4473 grams.
What is the slope of the line through (-9,6) and (-3, 9)?
Answers
9-6/-3-(-9)=
3/6= 1/2
Slope = 1/2.
Y=1/2x.
How would you type the answer for GE as in picture for an edpuzzle answer ? What does the check mark symbol mean ?
Answers
Based on the image attached, the solution for GE is approximately 22.72.
What is the expression ?To be able to solve the equation √129 + √129 = GE, one need to simplify the expression and then isolate GE.
Step 1: One can add the square roots:
Note that the equation:
√129 + √129 = GE.
Step 2: Do put together the square root terms:
2√129 = GE.
Step 3: Then Divide by 2:
√129 = GE/2.
Step 4: The Square both sides:
129 = (GE²)/4.
Step 5: So, Multiply by 4:
516 = GE².
Lastly, Take the square root:
√516 = √(GE²)
√516 = GE
GE = 22.72. approx.
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See text below
Step 26
Solve the equation for GE:
√129+√129 = GE
GE=2/129
Show steps
Rose and Angel are caterers. Both Rose and Angel charge a flat fee plus a fee per guest. Rose charges a flat fee of $50 plus $12 per guest. The table shows how much Angel charges for different numbers of guests. Angel's Catering Charges Number of Guests Charge ($) 10 $180 15 $255 25 $405 30 $480 Part A Who charges the greater flat fee? O A. Rapse O B. Angel Part B Vhat is the difference in total charges between Angel and Rose for a party that includes 50 guests? Enter the answer in the box. Autohide sh Growth: Math 6+ GA 2015 Question ITU Shelf nositi
Answers
Answer:
a. Angel has a greater flat fee
b. Difference = $130
Step-by-step explanation:
Given
Rose:
\(Flat\ Fee = \$50\)
\(Rate = \$12\)
Angel
Guests --- Charges
10 -------- $180
15 -------- $255
25 -------- $405
30 -------- $480
Solving (a): Who charges the greater flat fee?
We have that the flat fee of Rose is:
\(Flat\ Fee = \$50\)
For Angel, we need to determine the equation of the given table
Start by calculating the slope (m) of the table
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Where x and y represent any two corresponding values of the guests and the charges.
\((x_1,y_1) = (10, 180)\)
\((x_2,y_2) = (30, 480)\)
So: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) becomes
\(m = \frac{480 - 180}{30 - 10}\)
\(m = \frac{300}{20}\)
\(m = 15\)
Next, we calculate the equation using:
\(y - y_1 = m(x - x_1)\)
Where
\(m = 15\)
\((x_1,y_1) = (10, 180)\)
\(y - 180 = 15(x - 10)\)
\(y - 180 = 15x - 150\)
Add 180 to both sides
\(y - 180 + 180 = 15x - 150 + 180\)
\(y= 15x + 30\)
From the equation above:
The slope = 15 --- This represents the hourly rate
and
y intercept = 30 --- This represents the flat fee
This is better represented as:
Angel
\(Flat\ Fee = \$30\)
\(Rate = \$15\)
and
Rose:
\(Flat\ Fee = \$50\)
\(Rate = \$12\)
By comparison, Angel has a greater flat fee
Solving (b): Difference between total charges of 50 guests for both caterers.
For angel, the equation is:
\(y= 15x + 30\)
and x = 50.
So:
\(y = 15 * 50 + 30\)
\(y = 750 + 30\)
\(y = 780\)
For Rose,
First, we need to determine the equation.
\(Flat\ Fee = \$50\)
\(Rate = \$12\)
The equation is:
\(y = Flat\ Fee + Rate * x\)
\(y = 50 + 12 * x\)
So, the total charges for 50 guests is:
\(y = 50 + 12 * 50\)
\(y = 50 + 600\)
\(y = 650\)
The difference is then calculates as:
\(Difference = 780 - 650\)
\(Difference = \$130\)
if you double a number and then add 36, you get 4 over 11 (4/11) of the original number,
what is the original number?
Answers
Answer:
The original number is -22
Step-by-step explanation:
We'll label our mystery number x.
2x + 36 = 4x/11
Multiply both sides by 11
4x = 22x + 396
Isolate x to one side (for this I subtract 4x from both sides, but you can also subtract 22x if you'd like)
0 = 18x + 396
Isolate x pt 2
18x = -396
Divide both sides by 18 to find your answer!
x = -22
Plug in to confirm
-44 + 36 = -88/11S
8 = 8
a 24 foot tall streetlight casts a shadow that is 18 feet long. how long of a shadow is cast by a nearby parking meter post that is 4 feet high?
Answers
9514 1404 393
Answer:
3 ft
Step-by-step explanation:
The post is 4/24 = 1/6 as tall as the streetlight, so we expect its shadow to be 1/6 as long as that of the streetlight:
(1/6)(18 ft) = 3 ft
The parking meter post shadow is 3 feet long.
can u pls help me with this question
Answers
Consider the triangle A(4,0), B(-2,4), C(0,6)
What will be the equation of the altitude BM?
Answers
Answer:
BM: y = (2/3) x + 16/3 with segment length of 2.77
Step-by-step explanation:
AC formula: m = (6-0)/(0-4) = -3/2
(y-0)/(x-4) = -3/2 y = (-3/2)x + 6 ... (1)
BM slope: BM⊥ AC m = 2/3
BM formula: (y-4) / (x- -2) = (y-4) / (x+2) = 2/3
y-4 = 2/3 x + 4/3
y = (2/3) x + 16/3 ... (2) -2≤x≤0.31
intercept of AC and BM (M) from (1) and (2): (-3/2)x + 6 = (2/3) x + 16/3
(13/6) x = 2/3 x = (2/3) / (13/6) = 4/13 ≈ 0.31
y = (2/3) (4/13) + (16/3) = (8/39) + (208/39) = 216/39 = 72/13 ≈ 5.54
M (4/13 , 72/13) or (0.31 , 5.54)
segment BM = √(4/13 - -2)² + (72/13 - 4)² = √1300/169 = 2.77
Marking as brainliest!!
Answers
8^2+6^2=100
Sqrt(100)=10
Answer:
10 ft
Step-by-step explanation:
Given the function f(x) = -2x - 5, determine f(-3)?
Answers
In order to determine f(-3), we just need to use the value of x = -3 in the equation of f(x) and determine its value.
So we have:
\(\begin{gathered} f(x)=-2x-5 \\ f(-3)=-2\cdot(-3)-5 \\ f(-3)=6-5 \\ f(-3)=1 \end{gathered}\)So the value of f(-3) is 1.
A coin is tossed twice. Let
E
be the event "the first toss shows heads" and
F
the event "the second toss shows heads".
(a) Are the events
E
and
F
independent?
Input Yes or No:
(b) Find the probability of showing heads on both tosses. Write your answer as a reduced fraction.
Answer:
Answers
Answer:
(a) Yes, E and F are independent events.
(b) P(E and F) = P(E)P(F) = (1/2)(1/2) = 1/4
Find the area of the triangle. around intermediate values to the nearest tenth. Use rounded values to calculate the next value. Found your final answer to the nearest tenth
Answers
The area of the triangle is 97.07 units².
Given is a triangle we need to find the area,
So, let the triangle be ABC and the altitude be AD.
Using the trigonometric ratios,
Sin 52° = AB / AD
Sin 52° = 9 / AD
AD = 9 / Sin 52°
AD = 11.42
Now,
Tan 26° = DC / AD
Tan 26° = DC / 11.42
DC = 5.56
Also,
Cos 26° = AD / AC
Cos 26° = 11.42 / AC
AC = 12.7
BC = 11.42 + 5.56
BC ≈ 17
Now, the sides are BC = 17, AC = 12.7 and AB = 9,
The area of the triangle = 1/2 x BC x AD = 1/2 x 17 x 11.42
= 5.71 x 17
= 97.07
Hence the area of the triangle is 97.07 units².
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How many years would it take a $6,000 to triple in value with a simple interest rate of 5%?
Answers
Answer:
it would take a total of 22.52 years
Answer:
40 years
Step-by-step explanation:
Find the missing angle and side measures of Delta*ABC , given that
m angle A = 50 deg , m angle C = 90 deg , and CB = 16
Answers
The missing angle is <B= 40 degree and missing side length is AB = 12.25 and AC = 19.068.
To find the missing angle and side measures of ΔABC, we can use the properties of a triangle.
Given:
∠A = 50°
∠C = 90°
CB = 16
We can start by finding the measure of ∠B:
∠A + ∠B + ∠C = 180° (Sum of angles in a triangle)
50° + ∠B + 90° = 180°
∠B + 140° = 180°
∠B = 180° - 140°
∠B = 40°
Now, using Sine law
CB/ sin A = AB / sin C
16 / sin 50 = AB / sin 90
16 / 0.766044 = AB
AB = 12.25
Again 12.25 = AC/ sin B
12.25 = AC / sin 40
AC = 19.068
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A simple random sample of 8 reaction times of NASCAR drivers is selected. The reaction times have a normal distribution. The sample mean is 1.24
sec with a standard deviation of 0.12 sec. Construct a 99% confidence interval for the population standard deviation
Answers
Answer:
99% confidence interval is 0.07 < σ < 0.32
Step-by-step explanation:
Given that;
standard deviation s = 0.12 sec
s² = 0.12² = 0.0144
degree of freedom DF = n - 1 = 8 - 1 = 7
99% confidence interval
∝ = 1 - 99% = 1 - 0.99 = 0.01
now, we find x² critical values for ∝/2 = 0.005 and 1 - ∝/2 = ( 1 - 0.005) = 0.995, df = 7
The Lower critical value \(X^{2}_{\frac{\alpha }{2}},7\) = 20.2777
The Upper critical value \(X^{2}_{1-{\frac{\alpha }{2}},7\) = 0.9893
Now, confidence interval is given by
√[ ( (n-1)×s² ) / ( \(X^{2}_{\frac{\alpha }{2}},7\) ) ] < σ < √[ ( (n-1)×s² ) / ( \(X^{2}_{1-{\frac{\alpha }{2}},7\) ) ]
so we substitute
√[ ( 7×0.0144 ) / ( 20.2777 ) ] < σ < √[ ( 7×0.0144 ) / ( 0.9893 ) ]
√[ ( 7×0.0144 ) / ( 20.2777 ) ] < σ < √[ ( 7×0.0144 ) / ( 0.9893 ) ]
√0.0049711 < σ < √0.10189
0.0705 < σ < 0.3192
Rounding to the nearest 2 decimal places
0.07 < σ < 0.32
Therefore; 99% confidence interval is 0.07 < σ < 0.32
24 is 16% of what can you guys help and show work
Answers
Answer:
150
Step-by-step explanation:
24/x=16/100
Using cross-multiplication, we get 16x=2400
When we divide both sides by 16,
we get x=150, so, our answer is 150
ASAP it’s due tomorrow ASAP
Answers
Answer:
D
Step-by-step explanation:
_ No_(Did_Not_Lie) Yes_(Lied)
Positive_test_results 10 45
Negative_test_results 34 12
The table below displays results from experiments with polygraph instruments. Find the positive predictive value for the test. That is, find the probability that the subject lied, given that the test yields a positive result.
Answers
The probability that the subject lied, given that the test yields a positive result is; 81.81%
How to interpret the probability from a table?
From the given table, we see that;
Number of people with a positive result who did not lie = 10
Number of people with a positive result who lied = 45
Total number of people with positive results = (10 + 45) = 55
Number of people with a negative result who did not lie = 34
Number of people with a negative result who lied = 12
Total number of people with negative results = (34 + 12) = 46
The probability that the subject lied, given that the test yields a positive result is;
Probability = 45/55 = 0.818 = 81.81%
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Find RS. Explain your reasoning.
6x + 5 9x - 4
RS=
Answers
The value of line RS is 23
How to determine the valueFrom the diagram shown, we have that line PS, line PQ, line QR and line RS form a square.
Note that the sides of a square are equal to each other.
The relationship is represented as;
Line PS = Line PQ = Line QR = Line RS
We have that;
Line PS = 6x - 5Line RS = 9x - 4Substitute the expressions
6x - 5 = 9x - 4
Now, collect like terms
6x - 9x = - 4 - 5
Subtract the like terms
-3x = -9
Make 'x' the subject of formula
-3x/-3 = -9/-3
x = 3
Then, RS = 9x - 4 = 9(3) - 4 = 27 - 2 = 23
Hence, the value is 23
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3/4x4/5 in simplest list for
Answers
Answer:
3/5
Step-by-step explanation:
3/4 * 4/5= 12/20= 3/5
Please help I am struggling bad with this question thank you all
Answers
b = speed of the boat in still water
c = speed of the current
when going Upstream, the boat is not really going "b" fast, is really going slower, is going "b - c", because the current is subtracting speed from it, likewise, when going Downstream the boat is not going "b" fast, is really going faster, is going "b + c", because the current is adding its speed to it.
Now, the boat goes Upstream 48 miles, so Downstream must be travelling the same 48 miles back.
\({\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ Upstream&48&b-c&3\\ Downstream&48&b+c&2 \end{array}\hspace{5em} \begin{cases} 48=(b-c)(3)\\\\ 48=(b+c)(2) \end{cases} \\\\[-0.35em] ~\dotfill\)
\(\stackrel{\textit{using the 1st equation}}{48=(b-c)3}\implies \cfrac{48}{3}=b-c\implies 16=b-c\implies \boxed{16+c=b} \\\\\\ \stackrel{\textit{using the 2nd equation}}{48=(b+c)2}\implies \cfrac{48}{2}=b+c\implies 24=b+c\implies \stackrel{\textit{substituting}}{24=(16+c)+c} \\\\\\ 24=16+2c\implies 8=2c\implies \cfrac{8}{4}=c\implies \boxed{2=c}~\hfill~\stackrel{ 16~~ + ~~2 }{\boxed{b=18}}\)