Answer:
speed in still water is 15 miles per hour; the speed of the current is 9 miles per hour
Step-by-step explanation:
This is a distance = rate times time problem.
We have to set up a table with the info we have and then take it from there.
d = r x t
downstream (w/ current)
upstream (against current)
We know the distance is 72 miles each way, so filling that in (I am also abbreviating downstream to d.s. and upstream to u.s.)
d = r x t
d.s 72
u.s 72
We also know that the trip with the current took 3 hours, and the trip back took 12. Filling that in:
d = r x t
d.s 72 = x 3
u.s 72 = x 12
Now we just need the rate values. But we don't have anything solid to put in there. We only know that WITH the current, the rate of the boat is faster; we also know that AGAINST the current, the rate of the boat is slower. So the rate with the current is whatever the rate in still water is + the rate of the current or r + c. The rate against the current is whatever the rate is in still water - the rate of the current or r - c. Those values will fit into the rate column:
d = r x t
d.s 72 = (r + c) x 3
u.s. 72 = (r - c) x 12
Since distance = rate * time, we set that equation up for each part of the trip. For the first part:
72 = 3(r + c) and
72 = 3r + 3c
For the second part:
72 = 12(r - c) and
72 = 12r - 12c
Since the rate of the boat in still water is going to be the same whether you are being pushed along or being pushed against by the current, I solved the first equation for r:
72 = 3r + 3c and
3r = 72 - 3c so
r = 24 - c
and subbed that in for r in the second equation to solve for the rate of the current:
72 = 12(24 - c) - 12c and
72 = 288 - 12c - 12c
Combining like terms and we have
-216 = -24c so
c = 9
Now we can go back up the rate in terms of current, r = 24 - c, and plug in 9 for c to solve for r:
r = 24 - 9 so
r = 15
The boat's speed in still water is 15 miles per hour, and the speed of the current is 9 miles per hour.
The boat's speed in still water is 15 miles per hour, and the speed of the current is 9 miles per hour, determined by setting up equations based on rate, time, and distance for both downstream and upstream travel.
The question involves determining the speed of a boat in still water and the speed of the current given the times it takes to travel a certain distance downstream and upstream. To solve the problem, we use the concept of rate, time, and distance, where distance equals rate multiplied by time (D = RT). We know the distance (72 miles), the time downstream (3 hours), and the time upstream (12 hours).
Let's define the speed of the boat in still water as 'b' and the speed of the current as 'c'. When the boat travels downstream, it moves with the current, so its effective speed is 'b + c'. Upstream, it moves against the current, so its effective speed is 'b - c'. Using the distances and times given, we can set up two equations:
1) Downstream: 72 = (b + c) * 3
2) Upstream: 72 = (b - c) * 12
By solving these simultaneous equations, we find out:
For downstream: b + c = 72/3 = 24 miles per hour.For upstream: b - c = 72/12 = 6 miles per hour.Adding these two equations gives us 2b = 30, meaning b = 15 miles per hour. By substituting b in any of the above equations, we find c = 9 miles per hour.
Therefore, the boat's speed in still water is 15 miles per hour, and the speed of the current is 9 miles per hour.
find the outlier in the data:46, 65, 38, 42, 45, 46, 48, 42. how does the outlier affect the mean? if necessary, round to the nearest tenth?
pls help lol
Answer:
65
Step-by-step explanation:
The outlier made the mean bigger/greater. Rounded to the nearest tenth it would be 40.8.
A rancher wants to fence in an area of 3,000,000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?
Answer:
shortest length of fence is 8485.2 ft
Step-by-step explanation:
Given data
area = 3,000,000 square feet
to find out
shortest length of fence
let length L and width is W
so area is L × W
W = 3 × [tex]10^{6}[/tex] /L ............1
2W = 6 × [tex]10^{6}[/tex] /L
rectangular field and then divide it in half
so fencing will be 3 × L + 2 × W
i.e. 3 L + 2W
fencing = 3 L + 6 × [tex]10^{6}[/tex] /L
fencing minimum = 3 L - 6 × [tex]10^{6}[/tex] /L²
fencing minimum length will be zero
3 L - 6 × [tex]10^{6}[/tex] /L² = 0
3 L² = 6 × [tex]10^{6}[/tex]
L² = 2 × [tex]10^{6}[/tex]
L = 1414.2
so from equation 1
W = 3 × [tex]10^{6}[/tex] /L
W = 3 × [tex]10^{6}[/tex] /1414.2
W = 2121.3
so fencing will be 3 L +2 W
so fencing = 3 × 1414.2 +2 × 2121.3
fencing = 4242.6 +4242.6
fencing = 8485.2
shortest length of fence is 8485.2 ft
The shortest length of fence the rancher can use is approximately 6104 feet. This is derived by setting up the area and perimeter equations, differentiating to find the minimum perimeter, and substituting the values back into the equation.
Explanation:This problem is a basic optimization problem in mathematics. Given that the area of the field to be fenced is 3,000,000 square feet, we can use the formula for the area of a rectangle, which is length multiplied by width. Since the rancher wants to divide the field in two with a fence running parallel to one side, the total amount of fencing will be two lengths and three widths.
Let's denote the length of the rectangle as 'l' and the width as 'w'. The area is thus l*w = 3,000,000. The perimeter is defined as 2*l + 3*w. Given that the area is fixed, w can be expressed in terms of l as 3,000,000/l.
Therefore, the perimeter becomes 2*l + 3*(3,000,000/l). The minimum fence length or perimeter occurs when the derivative of this equation is zero. By differentiating and setting the equation to zero, we get l=sqrt(1,500,000), approximately 1224.74 feet. Substituting this value into the equation for w gives us w also as 1224.74 feet.
The shortest length of fence that the rancher can use is thus 2*l + 3*w = 2*1224.74 + 3*1224.74 = 6103.7 feet.
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Suppose u and v are a basis for the two-dimensional vector space V. Prove that w = u+v and x = u-v is also a basis. Hint: If two vectors form a basis, then any vector in the space can be expressed as linear combinations of the two vectors. You know that u and v are a basis. Pick any vector, call it s, in the space and check that you can always do the same using w and x.
Answer with Step-by-step explanation:
We are given that u and v are a basis for the two dimensional vector space.
To prove that w=u+v and x=u-v is also a basis .
By using matrix we prove w and x are basis of vector space.
We make a matrix from w and x
[tex]\left[\begin{array}{cc}1&1\\1&-1\end{array}\right][/tex]
Apply operation
[tex] R_1\rightarrow R_1-R_2[/tex]
[tex]\left[\begin{array}{cc}0&2\\1&-1\end{array}\right][/tex]
Apply [tex] R_2\rightarrow R_2-+R_1[/tex]
[tex]\left[\begin{array}{cc}0&2\\1&1\end{array}\right][/tex]
Apply [tex]R_1\rightarrow \frac{1}{2}R_1[/tex]
[tex]\left[\begin{array}{cc}0&1\\1&1\end{array}\right][/tex]
Apply [tex]R_2\rightarrow R_2-R_1[/tex]
[tex]\left[\begin{array}{cc}0&1\\1&0\end{array}\right][/tex]
Rank is 2 .Therefore, row one and second row are linearly independent.
Hence, first and second row are linearly independent because, any row is not a linear combination of other row.
Therefore, w and x are formed basis of given vector space.
Find the equation for the line below
Answer:
y = (-4/3)(x +4) +4
Step-by-step explanation:
Between the given points, the "rise" is -8 and the "run" is 6. The rise/run ratio is then -8/6 = -4/3. So, a version of the point-slope equation can be written:
y = m(x -h) +k . . . . . . . line with slope m through point (h, k)
y = (-4/3)(x +4) +4
_____
Comment on the answer
The image asks for "an equation", which seems to allow for any equation format. The equation shown above can be put into other forms.
y -4 = -4/3(x +4) . . . . different point-slope form
y = -4/3x -4/3 . . . . . . slope-intercept form
4x +3y = -4 . . . . . . . . standard form
4x +3y +4 = 0 . . . . . . general form
x/(-1) +y/(-4/3) = 1 . . . . intercept form
A 28,000-gallon swimming pool is being drained using a pump that empties 700 gallons per hour. Which equation models this situation if g is the number of gallons remaining in the pool and t is the amount of time in hours the pool has been draining? 28,000 = –700t 28,000g = –700t g = 700t – 28,000 g = 28,000 – 700t
Answer:
g = 28,000 - 700t
Step-by-step explanation:
This solution reads, in words,
"the amount of water remaining in the pool is equal to 28,000 gallons minus 700 gallons per hour", which is what your situation is asking you. You start with 28,000 gallons and are pumping out (subtracting) 700 gallons per hour.
g is what remains
Answer: [tex]g=28000-700t[/tex]
Step-by-step explanation:
Given : A 28,000-gallon swimming pool is being drained using a pump that empties 700 gallons per hour.
i.e, Remaining gallons = 28,000- 700 × Number of hours.
Let g be the number of gallons remaining in the pool and t is the amount of time in hours the pool has been draining.
Then, the equation models this situation will be :-
[tex]g=28000-700t[/tex]
The Taieri Plain in New Zealand is 6 feet below sea level (or –6 feet). If you are standing 3 feet above sea level, is your elevation the opposite of the elevation of the plain? Describe the steps you would follow with a vertical number line to find the answer.
Answer:
no
Step-by-step explanation:
Opposite numbers are the same distance from 0 on the number line, but in opposite directions. Steps I would follow:
first, check the directions to make sure they are opposite, then read the digits, or count the tick marks or compare lengths with dividers or ruler or marks on a piece of paper to see if the distances are the same.3 is not the opposite of -6
__
6 is the opposite of -6
Answer:
Sample Response: Although 6 and 3 are on opposite sides of sea level, or 0, they are not opposites because they are unequal distances from 0.
Step-by-step explanation:
I got the answer correct
Name the similar triangles.
triangles CBA and FDE with angle D congruent to angle B and angle E congruent to angle A
A. ΔABC ~ ΔDEF
B. ΔABC ~ ΔEDF
C. ΔABC ~ ΔDFE
D. ΔABC ~ ΔFED
Answer:
ABC ~ FED
Step-by-step explanation:
A ~ F
B~E
C~D
If you're still confused, look at the angles. No lines match to A and F, 1 line matches to B and E, and 2 lines match to C and D.
The correct answer is ΔABC ~ ΔDEF, where the corresponding congruent angles are matched to affirm triangle similarity, making option A correct.
Explanation:To determine which triangles are similar, we need to match the angles from ΔCBA to ΔFDE. Since angle D is congruent to angle B and angle E is congruent to angle A, we need to ensure that the order of the vertices corresponds to the congruent angles in the similar triangles. Therefore, the correct order that names the similar triangles is ΔABC ~ ΔDEF, which makes option A correct. In ΔABC, angle A corresponds to angle E in ΔDEF, and angle B corresponds to angle D. This order maintains the sequence of corresponding congruent angles in both triangles.
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Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. Together they charged a total of $775
. What was the rate charged per hour by each mechanic if the sum of the two rates was $100 per hour?
Answer:
102 an hour
Step-by-step explanation:
all you do is add all of them together
The students in Class A and Class B were asked how many pets they each have. The dot plots below show the results. What is the difference between the ranges in the dot plots?
Answer:
2
Step-by-step explanation:
The range of class A is 5-0 = 5
The range of class B is 3-0 = 3
the difference between 3 and 5 is 2.
The mean can be calculated by adding all of the values and dividing that sum by the number of values you added. To take advantage of the way data is being collected, I am grouping the addends as, for example, 5(1), instead of writing out 1 + 1 + 1 + 1 + 1.
\A CLASS mean is:
B CLASS mean is:
Based on these findings, we can conclude that the mean for Mrs. Ramos' class is about 2 more than the mean for Mr. McIntyre's class.
Find the sum of the geometric series
48 + 120 + . . . + 1875
a. 3093
b. 7780.5
c. 1218
d. 24037.5
Answer:
a. 3093
Step-by-step explanation:
The missing two terms in the 5-term sequence are ... 300 + 750.
You can add up these 5 terms directly, or you can recognize that the sum will be more than the last term (not C), but cannot be more than double the last term (not B or D). The sum must be 3093, choice A.
_____
If you use the formula for the general term of the series, you can find the number of terms.
The common ratio is 120/48 = 2.5
The general term is ...
an = a1·r^(n -1)
Solving for n, we get ...
n = log(an/a1)/log(r) +1 = log(1875/48)/log(2.5) +1 = 5 . . . . the number of terms
Then the sum is given by ...
Sn = a1(r^n -1)/(r -1) . . . . . using a1=48, r=2.5, n=5
S5 = 48(2.5^5 -1)/(2.5 -1) = 48(96.65625/1.5)
S5 = 3093
_____
The sum of a geometric sequence with a common ratio of 2 is double the last term, less the first term. When the common ratio gets larger, the sum gets smaller than double the last term.
Sn = a1(r^n -1)/(r -1) = (a1·r^n -a1)/(r -1) = (r·an -a1)/(r -1)
Sn = (r/(r -1))an -a1/(r -1) . . . . . . verifies the above comment
In our case, this evaluates to ...
S5 = (2.5/1.5)(1875) -48/1.5 = (5/3)(1875) -(2/3)(48)
= 3125 -32 = 3093
Using the first and last terms this way, we only need the common ratio and don't need to know the number of terms.
{(1,0.5)(2,0.25)(3,0.125)(4,0.0625)} Which kind of model best describes the data set?
Answer:
exponential: y = 2^(-x)
Step-by-step explanation:
The y-differences are not constant, and the x-y products are not constant. For each increase in x, y is divided by 2, so this is a geometric sequence that can be described by an exponential model. The common factor is 1/2.
A simple way to write the model for this is ...
y = 2^(-x)
A small publishing company is planning to publish a new book. The production costs will include one-time fixed costs (such as editing) and variable costs (such as printing). There are two production methods it could use. With one method, the one-time fixed costs will total $22,427, and the variable costs will be $19.25 per book. With the other method, the one-time fixed costs will total $53,962 , and the variable costs will be $10.50 per book. For how many books produced will the costs from the two methods be the same?
NEED HELP QUICK
Answer:
3,604 books
Step-by-step explanation:
We have 2 situations: situation A and situation B. What we are looking for is the number of books that has situation A equal to situation B. So the game plan is to write the equations for A and B, set them equal to each other, and then solve for the unknown number of books, x.
Situation A: If each book costs 19.25 to produce and the number of books is x, we express that as 19.25x. The fixed cost to use that company, regardless of the number of books it produces for you, is 22,427. Which means it is going to charge you 22,427 whether you produce 1000000 books or no books at all. The equation for A is:
C(A) = 19.25x + 22,427
Situation B uses the exact same reasoning, with the cost of each book being 10.50x and the flat rate cost of 53,962. Therefore, the equation for B:
C(B) = 10.50x + 53,962
We need the number of books where A = B, so we set the equations equal to each other and solve for x:
19.25x + 22,427 = 10.50x + 53,962 so
8.75x = 31,535 and
x = 3604
The Pacific Ocean is divided into two sets of gyres, in the _____ and _____. A. East Pacific; West Pacific B. North Pacific; East Pacific C. South Pacific; West Pacific D. North Pacific; South Pacific
Answer:
North pacific and south pacific.
Step-by-step explanation:
D.
The Pacific Ocean's two major gyre systems are found in the North Pacific and South Pacific.
Explanation:The Pacific Ocean is divided into two sets of gyres, notably in the North Pacific and South Pacific. Gyres are large systems of circulating ocean currents; the gyres in the Pacific Ocean play significant roles in influencing climate and marine life. The North Pacific Gyre, encircling the area between Asia and North America, and the South Pacific Gyre, contained mostly between Australia, South America and Antarctica, are the two main gyre systems in the Pacific Ocean.
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*HEY YALL I NEED HELP ASAP PLEASE*
Given the vectors u = <−3, 7> and v = <5,1>, find -½ u
<-1, -4>
<-4, -1>
<3/2, -7/2>
<-3/2, 7/2>
Step-by-step explanation:
given that u=<-3,7>
-1/2u=<-1/2(-3),-1/2(7)>
-1/2u=<3/2,-7/2>
Find the area of the shaded region under the standard distribution curve.
A. 2.5000
B. 0.9452
C. 0.1841
D. 0.7611
Answer:
D. 0.7611
Step-by-step explanation:
The area is:
P(z<1.60) − P(z<-0.90)
Looking up the values in a z-score table:
0.9452 − 0.1841
0.7611
Answer:
D. 0.7611
Step-by-step explanation:
We have been given a graph of a normal standard distribution curve. We are asked to find the area of the shaded region under the given standard distribution curve.
The area of the shaded region under the standard distribution curve would be area of a z-score of 1.60 minus area of a z-score of [tex]-0.90[/tex] that is [tex]P(-0.90<z<1.60)=P(z<1.60)-P(z<-0.90)[/tex]
Using normal distribution table, we will get:
[tex]P(-0.90<z<1.60)=0.94520-0.18406[/tex]
[tex]P(-0.90<z<1.60)=0.76114[/tex]
Therefore, the shaded area under the curve is 0.7611 and option D is the correct choice.
When taking a 12 question multiple choice test, where each question has 3 possible answers, it would be unusual to get _____ or more questions correct by guessing alone.
Assuming pure chance, on a 12 question multiple choice test with 3 options per question, it would be unusual to get 5 or more questions correct by guessing alone.
Explanation:The question pertains to the probability of guessing correctly on a multiple-choice test with 3 options per question, assuming pure chance. On a 12-question test, the probability of guessing correctly on a single question is 1/3. Hence, for 12 questions, the expected number of correct guesses would be the total number of questions times the probability of getting each question right, which is 12 * 1/3 = 4. Therefore, it would be unusual to get 5 or more questions correct by guessing alone.
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In a 12-question multiple choice test with 3 possible answers for each, a student randomly guessing is expected to get roughly 4 correct answers. Getting 6 or more questions correct by random guessing would be considered unusual.
Explanation:When taking a 12-question multiple choice test where each question has 3 possible answers, the probability of getting a question correct by simply guessing is 1/3. Here, we want to calculate the unusual scenario of how many or more questions correct by simply guessing.
If we apply the principle of probability, under normal circumstances where guessing is purely random, the expected number of correct answers would be the total number of questions times the probability of getting a question correct. This equates to 12 × (1/3) which is 4 correct answers. This means on average, if the student were to guess all their answers, they are likely to get around 4 correct answers.
However, getting 6 or more questions right by guessing alone would be considered unusual due to the low probability of guessing correct answers repeatedly. Keep in mind, that these calculations hold if all the guesses are random and there is no elimination of wrong answers based on known information.
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PLEASE HELP!!!!!
Mario sold concessions during the hockey tournament at the high school. He sold bottled water for $2.50 each and hamburgers for $5.50 each. By the end of the tournament, Mario’s sold 350 items for a total of $1,325. Which statements about the concession sales are true? Check all that apply.
A-The variable x could represent the number of bottles of water sold.
B-The variable x could represent the total sales amount for the bottled water.
C-The variable y could represent the total sales amount for the hamburgers.
D-The variable y could represent the number of hamburgers sold.
E-The system of equations that could represent the concession sales is
2.50x + 5.50y = 1,325
x+y=350
F-The system of equations that could represent this situation is
x+y=1,325
Answer:
A, D, E (probably preferred)
B, C, F (also true)
Step-by-step explanation:
"This situation" problems are almost always ambiguous. The first problem is to figure out what it is about "this situation" that you want to model.
__
If we want to model concession stand revenue in a general way, then statements B, C, and F could represent "this situation." No specific values can be determined from this model.
__
More likely, we want to model the details of sales and revenue, so we want to know the exact numbers of water bottles and hamburgers that were sold. For "this situation," we can assign variables to the numbers of items of each type, and write equations for the total number of items and for the revenue their sales generates. Appropriate statements about this interpretation of the problem are those of A, D, and E:
A: x could represent bottles sold
D: y could represent hamburgers sold
E: The system of equations could be 2.50x +5.50y = 1325; x +y = 350.
_____
The solution to the last model is 200 bottles were sold for revenue of $500, and 150 hamburgers were sold for revenue of $825.
Answer:
A-The variable x could represent the number of bottles of water sold.
D-The variable y could represent the number of hamburgers sold.
E-The system of equations that could represent the concession sales is
2.50x + 5.50y = 1,325
x+y=350
Step-by-step explanation:
When doing a system of equations you need variables and constants, in this case it´s easy to know what are the constants, since they are given directly to you by the problem, the constan 1 is the cost of the bottled water, the cost of each hamburger, the number of items sold and the total sum of the sells.
The only variables that we have are the number of items sold, we are going to express them with X and Y, the cost of the number of bottles of water sold will be expressed with the letter X and the number of hamburgers sold will be expressed by the letter Y.
Proportions in Triangles (8)
Answer:
not really sure maybe 12
Answer:
8 / 12 = x / 18
8 * 18 / 12 = x
x = 12
Step-by-step explanation:
The most common source of income is employment.
true or false?
Answer:
The most common source of income is business ownership. A. True.
Step-by-step explanation:
The most common source of income is employment. Other sources of income may include investments, business profits, or government assistance.
Explanation:The statement "The most common source of income is employment" is true.
Employment is the main source of income for the majority of people worldwide. It refers to a person's work for which they receive financial compensation in the form of wages or salary. Other sources of income may include investments, business profits, or government assistance, but employment is the most common and reliable.
Examples of employment include working in various sectors such as healthcare, technology, education, manufacturing, and services.
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Find the sum of the vectors <7,−2> and <1,8>. Then find the magnitude and direction of the resultant vector. Round angles to the nearest degree and other values to the nearest tenth.
Answer:
The sum of the vectors is <8 , 6>
The magnitude of the resultant vector is 10
The direction of the resultant vector is 37°
The answer is the 1st answer: <8 , 6> ; 10 ; 37°
Step-by-step explanation:
* Lets explain how to solve the problem
∵ The first vector is <7 , -2>
∵ The second vector is <1 , 8>
∴ The sum of the 2 vectors = <7 , -2> + <1 , 8>
∴ Their sum = <7 + 1 , -2 + 8> = <8 , 6>
* The sum of the vectors is <8 , 6>
- The magnitude of the resultant vector = √(x² + y²)
∵ x = 8 and y = 6
∴ The magnitude of the resultant vector = √(8² + ²)
∴ The magnitude of the resultant vector = √(36 + 64) = √100 = 10
* The magnitude of the resultant vector is 10
- The direction of the vector = tan^-1 (y/x)
∵ x = 8 and y = 6
∴ The direction of the vector = tan^-1 (6/8) = 36.869 ≅ 37°
* The direction of the resultant vector is 37°
Answer:
The answer is A. ⟨8, 6⟩; 10; 37°
(PLEASE HELP)
Two trains leave the station at the same time, one heading west and the other east. The westbound train travels at 55 miles per hour. The eastbound train travels at 75 miles per hour. How long will it take for the two trains to be 208 miles apart?
Do not do any rounding.
Answer:
1.6 hours or 1 hour 36 minutes.
Step-by-step explanation:
The rate at which the trains are moving apart is 55 + 75 = 130 mph.
Speed = distance / time so:
130 = 208 / t
130t = 208
t = 1.6 hours.
A radioactive substance decays by x % each day. After 8 days half of the substance has decayed. Find the value of x. Give your answer to 1 decimal place.
Answer:
8.3
Step-by-step explanation:
Let Ao be the original amount and A the amount after t days.
Then we have the exponential function
A = Ao(1 - x)^t or
A/Ao = (1 - x)^t
When t = 8, A/Ao = 0.5
0.5 = (1 - x)^8
(0.5)^(1/8) = 1 - x
0.917 = 1 - x
x = 1 - 0.917 = 0.083 = 8.3 %
The substance decays by 8.3 % each day.
What are the foci of the hyperbola whose equation is (x-6)^2/16-(y+7)^2/9 = 1?
(1,−7) and (11,−7)
(2,−7) and (10,−7)
(6,−12) and (6,−2)
(6,−10) and (6,−4)
Answer: (1, -7) (11, -7)
Step-by-step explanation:
The foci of the hyperbola whose equation is (1,−7) and (11,−7).
What are the foci of a hyperbola?The hyperbola is horizontal, we will count 5 spaces left and right and plot the foci there.
We need to use the formula [tex]\rm c^ 2 =a^ 2 +b^ 2[/tex] to find c.
The given equation of the hyperbola is;
[tex]\rm \dfrac{(x-6)^2}{16}-\dfrac{(y+7)^2}{9}=1[/tex]
Here a^2 is 16 and b^2 =9.
Substitute all the values in the formula
[tex]\rm c^ 2 =a^ 2 +b^ 2\\\\\rm c^ 2 16+9\\\\\rm c^2=25\\\\c^2=5^2\\\\c=5[/tex]
The center of the hyperbola is (6, -7).
The foci of the hyperbola whose equation is;
( 6 +5 , -7), (6-5, -7)
(11, -7), (1, -7)
Hence, the foci of the hyperbola whose equation is (1,−7) and (11,−7).
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An airplane needs to head due north, but there is a wind blowing from the northeast at 50 km/hr. The plane flies at an airspeed of 700 km/hr, To end up due north, the pilot will need to fly the plane Preview degrees east of north.
To solve the problem of the airplane needing to head due north while compensating for the wind blowing from the northeast, we'll need to use vector addition to determine the angle at which the pilot needs to fly.
First, we'll represent the airplane's velocity and the wind's velocity as vectors. Since the wind is blowing from the northeast, it has both a southward component and a westward component relative to due north.
Let's define our axes such that the north direction is positive y, and east is positive x. The airplane needs to end up flying due north, which is along the positive y-axis.
We'll denote:
- \( V_p \) = Velocity of the plane with respect to the air (airspeed) = 700 km/hr (directed at an angle we need to find, east of north),
- \( V_w \) = Velocity of the wind = 50 km/hr (from the northeast, so it has equal components to the south and west, at a -45-degree angle to the axes),
- \( V_r \) = Resultant velocity of the plane with respect to the ground (due north).
To cancel out the wind's effect and ensure that the plane ends up flying due north, the eastward (x) component of the plane's velocity must be equal in magnitude and opposite in direction to the eastward component of the wind's velocity.
Given that the wind velocity \( V_w \) is at a -45-degree angle, its components can be found using trigonometry:
\[ V_{wx} = V_w \cos(-45^\circ) \]
\[ V_{wy} = V_w \sin(-45^\circ) \]
Since \( \cos(-45^\circ) = \sin(-45^\circ) = \frac{\sqrt{2}}{2} \), and we're looking for the eastward component, \( V_{wx} \) will be positive and \( V_{wy} \) negative because it's to the south:
\[ V_{wx} = 50 \cdot \frac{\sqrt{2}}{2} \]
\[ V_{wx} = 25\sqrt{2} \]
Similarly, \( V_{wy} = -25\sqrt{2} \) but this component doesn't factor into our calculation for the required angle, since it affects only the north/south direction.
Now, to counteract this eastward wind component, the eastward component of the plane's velocity \( V_{px} \) must be equal to \( -V_{wx} \):
\[ V_{px} = -V_{wx} \]
\[ V_{px} = -25\sqrt{2} \] km/hr
The plane's airspeed is \( V_p = 700 \) km/hr, so using trigonometry with \( V_{px} \) and \( V_p \), we can find the angle \( \theta \) east of north that the plane should head:
\[ \sin(\theta) = \frac{V_{px}}{V_p} \]
\[ \sin(\theta) = \frac{-25\sqrt{2}}{700} \]
\[ \sin(\theta) = \frac{-25\sqrt{2}}{700} \approx -0.05071 \]
Taking the inverse sine (arcsin) to solve for \( \theta \):
\[ \theta = \arcsin(-0.05071) \]
The value of \( \theta \) obtained will be negative, indicating the angle west of north, but we want the angle east of north, so we take the absolute value:
\[ |\theta| \approx \arcsin(0.05071) \]
Using a calculator to find \( \theta \) in degrees:
\[ |\theta| \approx 2.905 \] degrees
Therefore, the pilot will have to fly approximately 2.905 degrees east of north to compensate for the northeast wind and ensure the resultant path is due north.
The pilot needs to fly approximately 4.28 degrees east of north to compensate for the wind and reach due north.
To calculate the angle the plane needs to fly east of north to compensate for the wind, we can use trigonometry. Here are the steps:
Step 1:
Determine the components of the wind.
The wind blowing from the northeast forms a right triangle with its northward and eastward components. The eastward component can be found using the sine function:
[tex]\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).[/tex]
[tex]\[ \sin(\theta) = \frac{50}{700} \][/tex]
Step 2:
Solve for the angle [tex]\( \theta \).[/tex]
[tex]\[ \theta = \arcsin\left(\frac{50}{700}\right) \][/tex]
Step 3:
Calculate the angle in degrees.
[tex]\[ \theta \approx 4.28^\circ \][/tex]
So, the pilot needs to fly approximately [tex]\( 4.28^\circ \)[/tex] east of north to compensate for the wind and end up due north.
The name of the lesson is "Using Trigonometric Laws" The two non-parallel sides of an isosceles trapezoid are each 7 feet long. The longer of the two bases measures 22 feet long. The sum of the base angles is 140°. a. Find the length of the diagonal. b. Find the length of the shorter base. Round your answers to the nearest hundredth.
Answer:
diagonal is 20.68 ft; shorter base is 17.21 ft
Step-by-step explanation:
If base angles add up to 140, then each is 70 since this is an isosceles trapezoid. We can use the Law of Cosines now to find the length of the diagonal which I will call d:
[tex]d^2=7^2+22^2-2(7)(22)cos(70)[/tex] and
[tex]d^2=533-308cos(70)[/tex] and
[tex]d^2=533-105.3422041[/tex],
[tex]d^2=427.6577959[/tex]
so the length of the diagonal is
d = 20.68 ft
The length of the shorter base is a little more tedious. First drop an altitude from each of the upper vertices to the base measuring 22 ft. We have 2 smaller right triangles now, each with unknown height (we need to find this), each with unknown base measures (we need to find this, too!), each with a base angle of 70 degrees, and each with a hypotenuse of 7. We need only work with one of these since they are both the same.
We will first find h, the height of each of these right triangles. Using the sin ratio:
[tex]sin(70)=\frac{h}{7}[/tex] and
[tex]7 sin(70)=h[/tex] so
h = 6.577848
Now we can use that along with the hypotenuse measure to find the base measure, b:
[tex]7^2-6.577848^2=b^2[/tex],
[tex]b^2=5.731911137[/tex] so
b = 2.394141002
Because we have 2 identical small right triangles with base measures of 2.394141002 each, we can subtract 2(2.394141002) from 22 to get the measure of the shorter base, which I will call x:
22 - 2(2.394141002) = x so
x = 17.21 ft
Amanda's family has a swimming pool that is 4 feet deep in their backyard. If the diameter is nearly 25 feet, what is the circumference of the pool?
Express your answer to the nearest whole foot.
Answer:
79 ft
Step-by-step explanation:
The formula for the circumference of a circle is
C = 2πr. If it's the diameter that is known, then C = πd. We will use the latter formula to find the circumference of this pool:
C = π(25 ft) = 78.54 ft, approx., which, to the nearest foot, is 79 ft
Jasmine is saving to buy a bicycle. The amount she has saved is shown in the table. What is the function describes the amount A, in dollars, Jasmine has saved after t weeks?
Table
Weeks/Amount
1 / $30
2 / $45
3 / $60
4 / $75
5 / $90
6 / $105
Answer:
A = 15t +15
Step-by-step explanation:
The amounts have a common difference of $15, so that is apparently the amount Jasmine is saving each week. Week 1, however, is $15 more than $15×1. The function ...
A = 15t +15 . . . . dollars
seems to fit the data.
The summer reading list for your English class has twelve books, and the list for History has eight books. You need to read three books for English and two books for History. How many different sets of books can you read?
Answer:
Step-by-step explanation:
This is a combination problem from stats. We have a total of 12 English books from which you have to 3. The order in which you pick them doesn't matter, you only need to determine how many different combinations are available to you. This is the combination formula, then:
₁₂C₃ = [tex]\frac{12!}{3!(12-3)!}[/tex]
I'm just going to simplify the right side and leave off the left side til the end of the algebra because it's easier. The right side simplifies to
[tex]\frac{12*11*10*9!}{3*2*1*9!}[/tex]
The 9!'s cancel each other out, leaving you with
[tex]\frac{12*11*10}{3*2*1}=\frac{1320}{6}[/tex]
Therefore,
₁₂C₃ = 220 possible different combinations of English books from which to pick.
We'll do the same for History, which has a combination formula that looks like this:
₈C₂= [tex]\frac{8!}{2!(8-2)!}[/tex]
That right side expands to
[tex]\frac{8*7*6!}{2*1*6!}[/tex]
The 6!'s cancel each other out, leaving you with:
[tex]\frac{8*7}{2*1}=\frac{56}{2}[/tex]
Therefore,
₈C₂ = 28 possible different combinations of History books from which to pick.
You may or may not need to add those together to get the answer your teacher is looking for.
Suppose your grandfather earned a salary of $12,000 in 1964. If the CPI is 31 in 1964 and 219 in 2016, then the value of your grandfather's salary in 2016 dollars is approximately
The value of a salary of $12,000 in 1964 would be approximately $84,000 in 2016 dollars when adjusted for inflation as measured by the Consumer Price Index.
Explanation:To calculate the value of your grandfather's salary in 2016 dollars, the relative rates of Consumer Price Index (CPI) for both years need to be compared. The salary of $12,000 in 1964, adjusted for inflation through 2016, would be:
$12,000 * (219 / 31)
This formula indicates the change in CPI from 1964 to 2016 and multiplies it by the salary in 1964 dollars. Therefore, your grandfather's salary in 2016 is approximately $84,000 when accounting for inflation.
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Final answer:
The value of your grandfather's salary in 2016 dollars is approximately $84,687.10
Explanation:
The value of your grandfather's salary in 2016 dollars is approximately $84,687.10
. To calculate this, you need to adjust the 1964 salary for inflation using the Consumer Price Index (CPI) for 1964 and 2016. First, find the inflation factor by dividing the CPI of 2016 by the CPI of 1964. Then, multiply the 1964 salary by this factor to get the equivalent value in 2016 dollars.
Which equation shows a valid, practical step in solving
For this case we have the following equation:
[tex]\sqrt [4] {2x-8} + \sqrt [4] {2x + 8} = 0[/tex]
If we subtract both sides of the equation [tex]\sqrt [4] {2x + 8}[/tex] we have:
[tex]\sqrt [4] {2x-8} = - \sqrt [4] {2x + 8}[/tex]
To eliminate the radical we raise both sides of the equation to the fourth power:
[tex](\sqrt [4] {2x-8}) ^ 4 = (- \sqrt [4] {2x + 8}) ^ 4[/tex]
Answer:
Option D