Answer:
Equation of parabola in vertex form:
y = -5( x + 1)² + 3
Equation of parabola in standard form:
y = -5x² -10x - 2
Step-by-step explanation:
The vertex form equation for a parabola is
y = a(x - h)² + k
where (h, k) is the vertex and a is a constant
Given (h, k) = (- 1, 3)
h = -1 , k = 3
y = a( x - (-1) )² + 3
y = a(x + 1)² + 3
To find a, we know the parabola passes through point(- 2, - 2). Plug in x = -2, y = -2 and solve for a
- 2 = a( -2 + 1)² + 3
-2 = a(-1)² + 3
-2 = a · 1 + 3
-2 = a + 3
Subtract 3 from both sides:
-2 - 3 = a + 3 -3
-5 = a
or
a = -5
Equation of parabola in vertex form:
y = -5( x + 1)² + 3
The standard form of this parabola is y = ax² + bx + c
Expand (x + 1)² = x² + 2 · 1 · x + 1² = x² + 2x + 1
Therefore
y = -5( x + 1)² + 3 becomes
y = -5( x² + 2x + 1) + 3
y = -5x² -10x - 5 + 3
y = -5x² -10x - 2
How can I find the zero while factoring for the equation I circled?
Answer:
Below
Step-by-step explanation:
x^2 -x + 1 Use Quadratic Formula a = 1 b = -1 c = 1
to find zeroes 1/2 ± i sqrt(3) / 3
Sooo.... Not sure you could find it by factoring:
(x -1/2 +i sqrt(3) /2) (x - 1/2 - i sqrt (3)/2) would be hard to see !!
a bird has a 90-inch cage. How many ft is that?
Answer:
7.5 feet
Step-by-step explanation:
Divide the length value by 12 for your answer
90/12 = 7.5
There are 12 inches in a foot, so to convert inches to feet, you can divide the number of inches by 12.
Therefore, a 90-inch cage is equivalent to:
90 inches ÷ 12 inches/foot = 7.5 feet
So the bird cage is 7.5 ft.
f(x)=2x^3+3x^2+50x+75 find all zeros of polynomial functions
All zeros of polynomial functions -3/2 , 5i and -5i
What is zeros of polynomial ?Zeros of polynomial are the points where the polynomial equals zero on the whole. In simple words, we can say that zeros of polynomial are values of the variable such that the polynomial equals 0 at that point. Zeros of a polynomial are also referred to as the roots of the equation and are often designated as α, β, γ respectively. Some of the methods used to find the zeros of polynomial are grouping, factorization, and using algebraic expressions.
2x^3+3x^2+50x+75 equal to 0
2x^3+3x^2+50x+75 = 0
⇒ x^2 (2x + 3) + 25 (2x + 3) = 0
⇒ (2x + 3) (x^2 + 25 ) = =
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0
2x + 3 = 0
x^2 + 25 = 0
so, x = -3/2 and x = 5i and -5i
The final solution is all the values that make 2x^3+3x^2+50x+75 = 0
x = -3/2 , 5i and -5i
Hence, all zeros of polynomial functions -3/2 , 5i and -5i
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8 yd
Area =
15 yd²
Volume=
I need to know how to solve and work it
The volume of the rectangle is 120 cubic yards
How to determine the volumeIt is important to note that the formula for calculating the volume of a rectangle is expressed as;
V = lwh
Given that the parameters are;
V is the volume of the rectanglel is the length of the rectangleh is the height of the rectanglew is the width of the rectangleFrom the information given, we that;
Area = 15 square yards
That is, length and width
Then,
Volume = 15(8)
multiply the values
Volume = 120 cubic yards
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The scale factor of two similar cylinders is 5:2.
The volume of the smaller cylinder is 28 m3.
What is the volume of the larger cylinder?
Question 8 options:
175 m3
350 m3
437.5 m3
70 m3
700 m3
Answer:
70m3
Step-by-step explanation:
5:2
The 2 represents the smaller cylinder. The smaller cylinder=28
To find out how much 1 is we can divide 28 by 2 and that =14. So in the ratio 1=14 and as seen on the left side we have 5 1's. So 14 times 5= 70
Have a nice day :-)
5) A decade of inflation
a) Suppose the annual inflation rate was 3% for 10 years in a row. What was the inflation rate for that decade?
b) Find the inflation rate for the decade from 1960 to 1970. If the inflation rate had been the same for each year in that decade, what would that annual rate have been? The answer to part a) will help you with this problem. This might be difficult depending on your math background, but try to be creative.
The inflation rate for that decade will be 34.39%. If the inflation rate had been the same for each year in that decade, the annual rate would have been 2.79%.
a) If the annual inflation rate was 3% for 10 years in a row, the inflation rate for that decade would be 34.39%. This is calculated by using the formula (1 + annual inflation rate)^number of years - 1. So in this case, it would be (1 + 0.03)^10 - 1 = 0.3439 or 34.39%.
b) To find the inflation rate for the decade from 1960 to 1970, we need to know the Consumer Price Index (CPI) for both years. The CPI for 1960 was 29.6 and the CPI for 1970 was 38.8.
We can use the formula (CPI in later year/CPI in earlier year)^(1/number of years) - 1 to find the annual inflation rate.
So in this case, it would be (38.8/29.6)^(1/10) - 1 = 0.0279 or 2.79%. Therefore, if the inflation rate had been the same for each year in that decade, the annual rate would have been 2.79%.
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The following TI-
84
Plus display presents some sample statistics.
1-Var-Stats
x
=67
Σ
x=2291
Σx2
=182,361
Sx=8
σx=8.024961059
↓n=29
Part: 0 /
In this case, the mean of the data set is 67, the sum of all the data values is 2291, the sum of all the squared data values is 182,361, the sample standard deviation is 8, the population standard deviation is 8.024961059, and the sample size is 29.
The TI-84 Plus display presents sample statistics for a set of data. The "x" symbol represents the mean, or average, of the data set. The "Σx" symbol represents the sum of all the data values, and the "Σx2" symbol represents the sum of all the squared data values.
These statistics can be used to analyze and interpret the data set. For example, the mean can be used to determine the central tendency of the data, and the standard deviation can be used to measure the spread of the data.
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59. Standard Normal areas Find the proportion of observations in a standard Normal distribution that satisfies each of the following statements.
(a) z > -1.66
(b) -1.66
(a) The proportion of observations in a standard normal distribution that satisfies z > -1.66 is 0.0475, or approximately 4.75%.
(b) The proportion of observations in a standard normal distribution that satisfies -1.66 < z < 1.66 is 0.9050, or approximately 90.50%.
What is z-score?A Z-score is a metric that quantifies how closely a value relates to the mean of a set of values.
Standard deviations from the mean are used to measure Z-score. A Z-score of zero means the data point's score is the same as the mean score.
To find the proportion of observations in a standard normal distribution that satisfies each of the statements, we need to use a standard normal distribution table or calculator.
(a) z > -1.66
We need to find the area to the right of z = -1.66 on a standard normal distribution table or calculator. This is equivalent to finding the area to the left of z = 1.66 since the standard normal distribution is symmetrical.
Using a standard normal distribution table or calculator, we find that the area to the left of z = 1.66 is 0.9525. Therefore, the area to the right of z = -1.66 is:
1 - 0.9525 = 0.0475
So, the proportion of observations in a standard normal distribution that satisfies z > -1.66 is 0.0475, or approximately 4.75%.
(b) -1.66 < z < 1.66
We need to find the area between z = -1.66 and z = 1.66 on a standard normal distribution table or calculator.
Using a standard normal distribution table or calculator, we find that the area to the left of z = 1.66 is 0.9525 and the area to the left of z = -1.66 is 0.0475 (as calculated in part (a)). Therefore, the area between z = -1.66 and z = 1.66 is:
0.9525 - 0.0475 = 0.9050
Therefore, the proportion of observations in a standard normal distribution that satisfies -1.66 < z < 1.66 is 0.9050, or approximately 90.50%.
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Covert 45km/h into m/s
Work Shown:
[tex]45\text{ km}/\text{hr} = \frac{45\text{ km}}{1\text{ hr}}\\\\=\frac{45\text{ km}}{1\text{ hr}}*\frac{1000\text{ m}}{1\text{ km}}*\frac{1\text{ hr}}{60\text{ min}}*\frac{1\text{ min}}{60\text{ sec}}\\\\=\frac{45*1000*1\text{ m}}{1*1*60*60\text{ sec}}\\\\=\frac{45000\text{ m}}{3600\text{ sec}}\\\\=12.5 \text{ m}/\text{s}\\\\[/tex]
Pay close attention to how the conversion fractions are set up on the second line. This placement is done specifically to have the units of "km", "hr", and "min" cancel out. The goal is to be left with "meters" up top and "seconds" down below.
What is the ratio of perimeters for two regular pentagons with areas of 18 cm² and 50 cm²?
18:50
09:25
15:25
O 3:5
Answer:
Step-by-step explanation:its 18.50
you buy an nft hoping it will increase in value at a compounded rate of 8% each year. if you pay $500 for the nft now. how much do you expect it to be worth in 20 years?
Using cοmpοund interest , the tοtal wοrth after 20 years is $2,330.48.
What is cοmpοund interest?The interest that is calculated using bοth the principal and the interest that has accrued during the previοus periοd is called cοmpοund interest. It differs frοm simple interest in that the principal is nοt taken intο accοunt when determining the interest fοr the subsequent periοd with simple interest.
Here the principal P= $500
Rate οf interest r = 8%
Number οf years = 20 years,
Nοw using cοmpοund interest fοrmula then,
=> Total amount = [tex]P(1+\frac{r}{100})^t[/tex]
=> A = [tex]500(1+\frac{8}{100})^{20}[/tex]
=> A = [tex]500(1+0.08)^{20}[/tex]
=> A = [tex]500(1.08)^{20}[/tex]
=>A = $2,330.48
Hence after 20 years , nft's total worth is $2,330.48.
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Find all the values of b that will make the trinomial 3x^(2)-bx+12 factorable? Choose one value of b and factor the resulting trinomial.
To find all the values of b that will make the trinomial 3x^(2)-bx+12 factorable, we need to use the discriminant of the quadratic formula. The discriminant is the part of the quadratic formula under the square root: b^(2)-4ac. If the discriminant is a perfect square, then the trinomial will be factorable.
So we plug in the values of a, b, and c from the trinomial: b^(2)-4(3)(12) = b^(2)-144.
We want this to be a perfect square, so we can set it equal to a perfect square and solve for b:
b^(2)-144 = 36
b^(2) = 180
b = sqrt(180)
b = 6sqrt(5)
So one value of b that will make the trinomial factorable is 6sqrt(5).
Now we can plug this value of b back into the trinomial and factor it:
3x^(2)-6sqrt(5)x+12 = 0
(3x-6)(x-sqrt(5)) = 0
So the factors of the trinomial are (3x-6) and (x-sqrt(5)).
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7. Parallelogram CDEF has vertices C(3, 2), D(8, 4), E(6, -3) and F(1, -5). Its image has vertices
C'(-6, 6), D'(-1, 8), E’(-3, 1), and
F’(-8, -1). Write a rule to represent this translation.
A rule to represent this translation include the following (x, y) → (x - 9, y + 4).
What is a translation?In Mathematics, the translation of a geometric figure to the right simply means adding a digit to the value on the x-coordinate (x-axis) of the pre-image of a function while a geometric figure that is translated upward simply means adding a digit to the value on the y-coordinate (y-axis) of the pre-image.
In Mathematics, a vertical translation to the positive y-direction (upward) is modeled by this mathematical expression g(x) = f(x) + N.
Where:
N represents an integer.g(x) and f(x) represent a function.C(3, 2) → C'(-6, 6)
3 + x = -6
x = -6 - 3 = -9.
y + 2 = 6
y = 6 - 2 = 4.
Therefore, the transformation rule is (x, y) → (x - 9, y + 4)
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enesis has a part-time job at an ice skating rink selling hot cocoa. She decided to plot the number of hot cocoas she sold relative to the day's high temperature and then draw the line of best fit. Based on the line of best fit, what would you predict the day’s high temperature to be if Genesis sold 28 hot cocoas?
based on the line of best fit, we would predict that the day's high temperature would be approximately 34.29 degrees Fahrenheit or 1.272°C if Genesis sold 28 hot cocoas
what is temperature?The term "temperature" describes how hot or cold a body is. It is, specifically, a method of calculating the kinetic energy of the particles that make up an item. When particles move more quickly, the temperature rises, and vice versa.
Nearly every aspect of our daily lives and all scientific fields, from physics to geology, are significantly influenced by temperature.
The velocity of these particles likewise increases as the temperature rises.
A thermometer or a calorimeter is used to determine the temperature. In other words, a system's internal energy is determined by its temperature.
from the question...
We may infer from the scatter plot and the line of best fit that the line's equation is y = -0.7x + 52, where y stands for the volume of hot cocoa sold and x for the high temperature of the day in degrees Fahrenheit.
If Genesis sold 28 hot cocoas, we can solve for x and substitute y = 28 into the equation to determine the day's high temperature:
28 = -0.7x + 52
-0.7x = 28 - 52
-0.7x = -24
x = 34.29
So, if Genesis sold 28 hot cocoas, we would forecast that the day's high temperature would be roughly 34.29 degrees Fahrenheit based on the line of best fit.
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complete question:
What statement describes the decimal equivalent of 5 ? 16 Responses A It is a decimal that terminates after 4 decimal places. It is a decimal that terminates after 4 decimal places. B It is a decimal with a repeating digit of 5. It is a decimal with a repeating digit of 5. C It is a decimal that terminates after 3 decimal places. It is a decimal that terminates after 3 decimal places. D It is a decimal with repeating digits of 6
The statement which describes the decimal equivalent of 5/16 is option(a), It is a decimal that terminates after 4 decimal places.
To convert a fraction to a decimal, we need to divide the numerator by the denominator.
In this case, we want to find the decimal equivalent of the fraction 5/16,
⇒ 5 ÷ 16 = 0.3125
So, the decimal equivalent of 5/16 is 0.3125.
The option(a) states that, It is a decimal that terminates after 4 decimal places.
This statement is correct, because the decimal equivalent of 5/16 terminates after 4 decimal places.
Therefore, the correct statement is option is (a).
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The given question is incomplete, the complete question is
What statement describes the decimal equivalent of 5/16?
(a) It is a decimal that terminates after 4 decimal places.
(b) It is a decimal with a repeating digit of 5.
(c) It is a decimal that terminates after 3 decimal places.
(d) It is a decimal with repeating digits of 6.
te the following polynomial operations and simplify the r (7x^(3)+3x^(4)-x^(5)+12)+(-13x^(5)+2x^(4)-4x^(2)+x+5).
The simplified result of the polynomial operations is: -14x^5 + 5x^4 + 7x^3 - 4x^2 + x + 17
To complete the polynomial operations and simplify the result, we need to combine like terms. Like terms are terms that have the same variable and the same exponent.
First, let's rewrite the expression to make it easier to see the like terms:
(7x^3 + 3x^4 - x^5 + 12) + (-13x^5 + 2x^4 - 4x^2 + x + 5)
Next, let's combine the like terms:
7x^3 + 3x^4 + 2x^4 - x^5 - 13x^5 - 4x^2 + x + 12 + 5
Simplify the expression by adding or subtracting the coefficients of the like terms:
5x^4 + 7x^3 - 14x^5 - 4x^2 + x + 17
The simplified result of the polynomial operations is:
-14x^5 + 5x^4 + 7x^3 - 4x^2 + x + 17
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Proving triangles congruent by ASA and AAS
The complete proof that ΔUWX ≅ ΔWUV is explained below.
What are congruent triangles?Congruent triangles are set of given triangles which have equal values of corresponding properties. Thus the triangles have equal dimensions and measure of internal angles.
The two column complete proof required are given below:
STATEMENT REASONS
1. WX ║ UV Given
2. < V ≅ <X Given
3. <VUW ≅ <UWX Definition of alternate angles.
4. UW ≅ UW Reflexive property of congruence
5. ΔUWX ≅ ΔWUV Angle-Side-Angle (AAS) property
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Find the perimeter of the triangle. Please show your work.
Answer:
Perimeter=33
Step-by-step explanation:
2x+3=4x-3
0=2x-8
2x=8
x=4
Input 4 as Variable
2x+3=2(4)+3
=8+3
=11
P=3x11
P=33
Find a function of the form y= A sin(kx) + C or y= A cos(kx) + C whose graph matches the function
shown below:
The equation of the sinusoidal graph will be y = 3 sin[(2/π)x] - 3.
What is trigonometry?Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.
The graph is represented by the sinusoidal equation.
From the graph, the amplitude of the graph is 3 units and the graph is shifted 3 units down.
And the value of constant k is given as,
k = 4/2π
k = 2/π
Thus, the equation of the sinusoidal graph will be y = 3 sin[(2/π)x] - 3.
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You told your friend that you could eat
of a large pizza. If you only ate
of what you said you could eat, what fraction of a large pizza did you eat?
I told my friend that I can eat 58 of a large pizza. If I only ate 12 of what I said, I can eat, then the fraction of large pizza eaten is [tex]\frac{6}{29}[/tex]
The portion of pizza supposed to be eaten was 58 while actually eaten slices of pizza is 12. The computation of the fraction of large pizza did you eat is given below:
= Ate portion ÷ large pizza
= [tex]\frac{12}{58}[/tex]
= [tex]\frac{6}{29}[/tex]
Hence, the fraction of large pizza did you eat is 6 ÷ 29
The question is incomplete the complete question would be "You told your friend that you could eat 58 of a large pizza. If you only ate 12 of what you said you could eat, what fraction of a large pizza did you eat?"
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what number is close to 48/100 is it 0 or 1 or 1/2?
Answer:
1/2
Step-by-step explanation:
Our answer options are: 0, 1, 1/2
0 = 0/100
1 = 100/100
1/2 = 50/100
Comparing the 3 options, we see that 1/2 is the closest to 48/100!
So, the answer is 1/2
Darrell wants to see how much water is wasted by his leaky faucet. He put a 4 gallon bucket under the faucet. After 24 hours, the bucket was full.
The amount of water filled in bucket 6 hours is found as 1 gallon.
Explain about the proportion of the number?Mathematical proportions are comparisons of two numbers that categorize or persons. They are frequently expressed as fractions or with a colon. There are two ways to write mathematical proportions. You can use colons to compare the numbers, or you can use equivalent fractions to represent the proportion.
Darrell is interested in learning how much water his leaking faucet is wasting. Under the faucet, he positioned a 4 gallon bucket. The bucket was full after 24 hours.Now,
Let the amount of water filled in 6 hours be 'x'.
Using proportion:
4/24 = x/6
x = 4*6 / 24
x = 1 gallon
Thus, the amount of water filled in bucket 6 hours is found as 1 gallon.
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The complete question is attached.
¡A survey was conducted at an amusement park to determine the preferred activity of the patrons. Each person chose one activity. The results are shown in the table. Activity Roller coasters Shows Food Number of Patrons 36 12 18 Based on this information, which prediction about the preferred activity for the next 300 patrons is most reasonable? A The number of patrons who prefer shows will be 5 times the number of patrons who prefer food. B The number of patrons who prefer food will be 2 times the number of patrons who prefer shows. C The number of patrons who prefer roller coasters will be 4 times the number of patrons who prefer food. D The number of patrons who prefer roller coasters will be 2 times the number of patrons who prefer food.
Option D: The number of patrons who prefer roller coasters will be 2 times the number of patrons who prefer food.
How to obtain the proportions?The activities are given as follows:
Roller coasters.Shows.Foods.The number of patrons for each activity is given as follows:
Roller coasters: 36.Shows: 12.Foods: 18.Hence the proportions are:
Roller coasters: 36/66.Shows: 12/66.Foods: 18/66.For the next 300 patrons, the estimates are given as follows:
Roller coasters: 36/66 x 300 = 164.Shows: 12/66 x 300 = 54.5.Foods: 18/66 = 82.Hence option d is correct.
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√(2+4)²-(-6+ (-4) ²
Loni wants to open a nail salon. Loni figured out that she can average charging $30 per session. Each session she does will cost her $10 in supplies. Over the first year, she has projected $24,000 in rent and $50,000 in other expenses. The business cost her $15,000to start. How many sessions does she need to do in the first year to break even?
Loni needs to do 4,450 sessions in the first year to break even.
What is total revenue?
Total revenue is the total amount of money earned by a business from the sale of its products or services during a particular period of time.
To break even, the total revenue Loni makes should be equal to the total cost of running the nail salon. Let's calculate the total cost first:
Total Cost = Rent + Other expenses + Start-up cost + Cost of supplies per session x Number of sessions
Total Cost = $24,000 + $50,000 + $15,000 + $10 x Number of sessions
Total Cost = $89,000 + $10 x Number of sessions
Now, to break even, the total revenue should be equal to the total cost:
Total Revenue = Total Cost
$30 x Number of sessions = $89,000 + $10 x Number of sessions
$20 x Number of sessions = $89,000
Number of sessions = $89,000 / $20
Number of sessions = 4,450
Therefore, Loni needs to do 4,450 sessions in the first year to break even.
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Eddie is laying out a design for the tiles on a section of his bathroom floor. He will use 1 black tile for every 4 white tiles to have total of 25 tiles in the design. How many white tiles will Eddie use?
Answer: :)
Step-by-step explanation:
Let's start by setting up a proportion to represent the ratio of black tiles to white tiles:
1 black tile : 4 white tiles
We know that Eddie wants a total of 25 tiles in the design, so we can set up another proportion to represent the ratio of white tiles to the total number of tiles:
white tiles : 25 total tiles
To solve for the number of white tiles, we need to find the equivalent ratio of white tiles in terms of the total number of tiles. We can do this by cross-multiplying our two proportions:
1 black tile * (white tiles/4 black tiles) = white tiles/4
white tiles/25 total tiles = (white tiles/4) / (1 black tile + white tiles/4)
Simplifying this expression, we get:
white tiles/25 = (white tiles/4) / (5/4)
white tiles/25 = (1/4) * (white tiles/5)
5 * white tiles = 100
white tiles = 20
Therefore, Eddie will use 20 white tiles in his design.
A. Create two subsets of the state population data: one with 2018 population greater than or equal to 5 million and one with 2018 population less than 5 million. How many observations are in each subset?
The probability of first subset has 14 observations with population of 5 million or greater, and the second subset has 36 observations with population less than 5 million.
The state population data has 50 observations, each representing a different state. To create two subsets, I filtered the data according to 2018 population. The first subset includes observations with population of 5 million or greater, and the second subset includes observations with population less than 5 million. There are 14 observations in the first subset and 36 observations in the second subset. This shows that there are more states with population below 5 million than above 5 million. The data also shows that the states with population of 5 million or greater tend to have higher population densities than those with population below 5 million. This could be due to a variety of factors such as location, climate, and resources.
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Help me please
Can someone answer number 4 please?
Will give brainlest
The measures of angles 4 and 5 are:
∠5 = 104°
∠4 =76°
How to find the measures of the angles?On the image we can see that angles 1 and 2 are next to eachother, that means that the sum of their measures must be a plane angle, that is an angle of 180°.
Then we can write the sum:
∠1 + ∠2 = 180°
(3x + 5) + (2x + 10) = 180
5x + 15 = 180
5x = 180 - 15
x = 165/5 = 33
the measure of angle 5 is the same one of angle 1 then:
∠5 = (3*33 + 5)° = 104°
And angle 4 is supplementary of angle 1, then:
∠4 + 104° = 180°
∠4 = 180 - 104= 76°
These are the measures
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An angle measures 6° more than the measure of its supplementary angle. What is the measure of each angle?
The two supplementary angles where one angle measures 6° more than the other one are 87° and 93°.
What are supplementary angles?
Two angles are said to be supplementary if the sum of the two angles is 180 degrees. One way to avoid confusion in these definitions is to note that s comes after c in the alphabet and 180 is greater than 90.
Solution according to the information in the question:
Let, "x" be one of the two supplementary angles
then, the other angle will be "x+6" according to the question.
As we know sum of two supplementary angles is 180 degrees, so
x +(x+6)=180
⇒2x + 6 = 180
⇒2x = 180 - 6
⇒2x = 174
⇒x = 174/2
⇒x = 87°
∴ The other angle = x +6 = 87 +6 = 93°
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The measure of the angle being discussed is 84° and the measure of its supplementary angle is 96°. This is because the measure of the angle being discussed is 6° more than the measure of its supplementary angle.
What is a supplementary angle?A supplementary angle is two angles whose sum is 180 degrees. These angles do not have to be adjacent or even in the same plane, but their sum must always equal 180 degrees.
If the angle being discussed is 6° more than the measure of its supplementary angle, then the measure of the angle being discussed is 84° (180° - 6° = 84°). Therefore, the measure of the angle being discussed is 84° and the measure of its supplementary angle is 180° - 84° = 96°.
To further explain the relationship between the two angles, the supplementary angle is the measure of the angle that completes the 180° angle when combined with the angle being discussed. This means that the measure of the supplementary angle is the 180° minus the measure of the angle being discussed. In this case, the measure of the angle being discussed is 84° and thus, the measure of its supplementary angle is 180° - 84° = 96°.
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A motor boat took 5 h to travel a distance of
60 km up a river, against the current. The return trip took 3 h. Find the average speed of the boat in still water and the speed of the current.
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.
[tex]{\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{km}{distance}&\stackrel{kmh}{rate}&\stackrel{hrs}{time}\\ \cline{2-4}&\\ Upstream&60&b-c&5\\ Downstream&60&b+c&3 \end{array}\hspace{5em} \begin{cases} 60=(b-c)(5)\\\\ 60=(b+c)(3) \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{using the 1st equation}}{60=(b-c)5}\implies \cfrac{60}{5}=bc\implies 12=b-c\implies 12+c=b \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 2nd equation}}{60=(b+c)3}\implies \cfrac{60}{3}=b+c\implies 20=b+c\implies \stackrel{\textit{substituting from above}}{20=(12+c)+c} \\\\\\ 20=12+2c\implies 8=2c\implies \cfrac{8}{2}=c\implies \boxed{4=c}~\hfill \stackrel{ 12~~ + ~~4 }{\boxed{b=16}}[/tex]
