Alice is driving eastward at 40mph, bob is moving northward at 60mph, and both are headed towards the same city. How fast is the distance between Alice and bob changing when they are both 10 miles from the city?
Answers
The given problem states that Alice is driving eastward at 40mph, Bob is moving northward at 60mph, and both are headed toward the same city.
We are asked to determine how fast the distance between Alice and Bob is changing when they are both 10 miles from the city.
Distance between Alice and Bob
The distance between Alice and Bob can be calculated using the Pythagorean theorem, as they are moving at right angles to each other. Let x be the distance of Alice from the city and y be the distance of Bob from the city.
Distance of Alice from the city
x = 10 miles distance of Bob from the city
y = 10 milesThe distance between Alice and Bob can be determined using the Pythagorean theorem.
Distance between Alice and Bob = √(x² + y²)
Let's substitute x = 10 miles and y = 10 miles in the above equation.
Distance between Alice and Bob
= √(10² + 10²)≈ 14. 14 miles
Differentiating with respect to time
We need to differentiate the equation for the distance between Alice and Bob with respect to time to find how fast it is changing over time.
Let's differentiate the above equation with respect to time.⇒ Distance between Alice and Bob = √(x² + y²)⇒ Differentiating both sides of the equation with respect to time, we get:
d/dt [Distance between Alice and Bob]
= d/dt[√(x² + y²)]⇒ d/dt[Distance between Alice and Bob]
= (1/2)[(x² + y²)^(-1/2)](2x dx/dt + 2y dy/dt)We know that dx/dt
= 40 mph (Alice is driving eastward) and dy/dt
= 60 mph (Bob is moving northward).
Let's substitute these values in the above equation to find how fast the distance between Alice and Bob is changing when they are both 10 miles from the city.
Distance between Alice and Bob
= √(10² + 10²) = ≈ 14.14 milesdx/dt
= 40 mphdy/dt
= 60 mphx
= 10 milesy
= 10 milesd/dt[Distance between Alice and Bob]
= (1/2)[(x² + y²)^(-1/2)](2x dx/dt + 2y dy/dt)⇒
d/dt[Distance between Alice and Bob]
= (1/2)[(10² + 10²)^(-1/2)](2 × 10 × 40 + 2 × 10 × 60)⇒ d/dt
[Distance between Alice and Bob] = 100√2 mph
Hence, the distance between Alice and Bob is changing at a rate of 100√2 mph when they are both 10 miles from the city.
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Related Questions
2(x + 4) < 4x - 6 smallest integer
Answers
Answer:
The smallest integer is 8
Step-by-step explanation:
2( x + 4 ) < 4x - 6
4x - 6 > 2x + 8
2x > 14
x > 7
The smallest integer is 8
Guys please help!!!!
If you have exponents in your answer please use ^ for example:
2^3= two to the power of three
Answers
Answer:
perimeter=6x+10
area=2x^2+2x-24
Step-by-step explanation:
perimeter=2x+8+x-3+2x+8+x-3
area=(2x+8)(x-3)
WORTH 20 POINTS If mABC = 250°, what is m∠ABC?
Answers
Answer:
55 degrees
Step-by-step explanation:
I've found a similar question to this, and the explanation is there.
https://brainly.com/question/14993673
m<ABC = 360-250= 110 degrees
"As we know that the measure of angle ABC is equal to half of mADC."
110/2 = 55 degrees.
This should be the answer.
The population of racoons in a state park increas by 3.5% each year. Right now, there are estimated to be 280 racoons in the park. How many raccoons will there be in 10 years
Answers
In the next 10 years there will be 398 racoons in the state park
What is population increase?The annual increase in the population size is defined as a sum of differences: the difference between births less deaths and the difference between immigrants less emigrants, in a given country, territory or geographic area at a given year.
Using the formula
p(t) = p(o) e^ kt
k = 3.5/100 = 0.035
t = 10 years
p(t) = 280 e^ 0.035 ×10
p(t) = 280e^0.35
p(t) = 280 × 1.42
p(t) = 398( nearest whole number)
therefore be after 10 years there will be 398 racoons in the state park.
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Do you think the following statement is true or false? "An atom is the smallest unit of matter. Atoms are the basic unit of an element, that are composed of protons, neutrons, and electrons."
Answers
The smallest unit of matter is an atom. The fundamental building block of an element is an atom, which is made up of protons, neutrons, and electrons. This statement is true.
An atom is what?An atom is made up of a nucleus and one or more electrons that are bound to the nucleus. The nucleus is made up of one or more protons, a significant number of neutrons, and other particles. Only the most common form of hydrogen lacks neutrons.
An atom, which is the smallest part of a material, retains all of its properties. A molecule is made up of protons, neutrons, and electrons, which are the building components of matter. Protons and neutrons, which are the only particles in an atom that contribute to its atomic mass, are found in the nucleus of each atom. These early 20th-century discoveries of particles that a nuclide itself and chemical components are explained by contemporary atomic theory.
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suppose only 48% of the students you selected for the sample respond. what is the nonresponse rate (in percent)?
Answers
The nonresponse rate is 52%.
The nonresponse rate refers to the percentage of individuals in a sample who do not provide a response or do not participate in the survey or study. In this case, if only 48% of the students selected for the sample respond, the remaining percentage represents the nonresponse rate.
To calculate the nonresponse rate, we subtract the response rate from 100%. In this case, the response rate is 48%, so the nonresponse rate can be calculated as:
Nonresponse rate = 100% - Response rate
= 100% - 48%
= 52%
The nonresponse rate is important to consider in any study or survey as it may introduce potential bias if the nonrespondents have different characteristics or opinions compared to the respondents. It is crucial to analyze and address the nonresponse bias to ensure the sample represents the target population accurately.
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please help me solve this fractions. (show your work)
Answers
Answer:
23: 21\(\frac{23}{55}\) 25: 1\(\frac{44}{45}\)
Step-by-step explanation:
Question 23:
1) add the whole numbers together: 15+5=20
2)To add fractions, both need to have the same denominator. Find a number that is a multiple of both denominators (in this case 55)
\(\frac{9}{11}\) → \(\frac{45}{55}\) (multiply the numerator and denominator by 5)
\(\frac{3}{5}\) → \(\frac{33}{55}\) (multiply the numerator and denominator by 11)
3) add the fractions together
\(\frac{45}{55}\) + \(\frac{33}{55}\) = \(\frac{78}{55}\) (add the numerators)
4) Turn the improper fraction into a mixed number
Divide 78 by 55. You'll get a whole number and a remainder. The remainder is the new numerator: In this case 55 enters 78 once with a remainder of 23. Therefore:
1\(\frac{23}{55}\)
5) Add the whole numbers together to get the final answer:
20+1= 21
Answer: 21\(\frac{23}{55}\)
Question 25:
Follow the above steps but subtract instead of adding
( 3,4) write the new coordinates after the following transformations. reflection over the x axis. reflection over the y axis. reflection over the line y=x
Answers
Let A be the point (3,4). Then, the first transformation is a reflection over te x-axis, which can be found using the following expression:
\(r_x(x,y)=(x,-y)\)in this case, we have:
\(r_x(A)=r_x(3,4)=(3,-4)=A_1\)the next transformation is a reflection over the y-axis, which has the following general rule:
\(r_y(x,y)=(-x,y)\)then, applying this transformation,we get:
\(r_y(A_1)=r_y(3,-4)=(-3,-4)=A_2\)finally, the reflection over the line y = x has the following function:
\(r_{y=x}(x,y)=(-x,-y)\)then, we have:
\(r_{y=x}(A_2)=r_{y=x}(-3,-4)=(-4,-3)\)therefore, the new coordinates after the transformations are (-4,-3)
please help me !!!!!!!!!!!!!!!!
Answers
5Step-by-step explanation:
Can someone help me with this question?
Answers
a + 5 is parallel to 15, meaning that
a + 5 = 15
a = 15 - 5
a = 10
3b° has the same value of 111°, so
3b = 111
b = 111 ÷ 3
b = 37
a = 10 , b = 37
Which one of the following is discrete data?
Answers
Answer:
Number of people in a room
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
if a is an nxn matrix that has n linearly independent eigenvectors, prove that also has n linearly independent eigenvectors.
Answers
To prove that a matrix A with n linearly independent eigenvectors also has n linearly independent eigenvectors, we need to use the fact that eigenvectors corresponding to distinct eigenvalues are linearly independent.
Let λ1, λ2, ..., λn be the n distinct eigenvalues of A with corresponding eigenvectors v1, v2, ..., vn, respectively. Since these eigenvectors are linearly independent, we know that any linear combination of them that equals the zero vector must have all coefficients equal to zero.
Suppose now that there exists a linear combination of these eigenvectors that equals the zero vector, say c1v1 + c2v2 + ... + cnvn = 0, where not all ci are zero. Without loss of generality, assume that c1 is nonzero. Multiplying both sides of this equation by A, we obtain
c1λ1v1 + c2λ2v2 + ... + cnλnvn = 0.
Since λ1, λ2, ..., λn are distinct, we know that λ1 cannot be equal to λ2, λ3, ..., λn. Therefore, we can divide both sides of the above equation by λ1 and obtain
c1v1 + (c2λ2/λ1)v2 + ... + (cnλn/λ1)vn = 0.
This equation shows that we have a nontrivial linear combination of the eigenvectors that equals the zero vector, contradicting the assumption that they are linearly independent. Therefore, our original assumption must be false, and we conclude that A has n linearly independent eigenvectors.
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How do you solve an infinite arithmetic sequence?
Answers
We have explained all the steps to solve an infinite arithmetic sequence.
What is an arithmetic sequence?
An arithmetic sequence is a sequence of numbers in which each term is the sum of the previous term and a constant called the common difference.
When it comes to solving an infinite arithmetic sequence, you can use the following formulas:
The n-th term of an arithmetic sequence can be represented as a+(n-1)d, where "a" is the first term, "d" is a common difference, and "n" is the position of the term in the sequence.
The sum of the first n terms of an arithmetic sequence is represented as S_n = n/2(2a + (n-1)d), where "a" is the first term, "d" is a common difference, and "n" is the number of terms.
If a sequence is infinite, the sum of all terms can be represented as S_inf = a/(1-r) where "a" is the first term and "r" is the common ratio (r=1+d)
Hence, we have explained all the steps to solve an infinite arithmetic sequence.
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How do you know if triangles are congruent in SAS?
Answers
With the help of the figure we can say that triangles are congruent by SAS Rule
What is Congruence of Triangle?
Triangle congruence: Two triangles are said to be congruent if all three of their corresponding sides are equal and all three of their corresponding angles are equal in size. These triangles can be moved, rotated, flipped, and turned to look exactly the same.
Solution:
By SAS rule, two triangles are said to be congruent if any two sides and the angle included between the sides of one triangle are comparable to the corresponding two sides and the angle included between the sides of the second triangle.
In given figure, sides AB= PQ, BC=QR and angle between AB and BC equal to angle between PQ and QR i.e. ∠B = ∠Q. Hence, Δ ABC ≅ Δ PQR.
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Friction creates heat and lowers the _______
Answers
Answer:
energy
Step-by-step explanation:
Answer:
energy
Step-by-step explanation:
For what values of x is the rational expression below undefined?
+5
32 - 3
Answers
Answer:
29
Step-by-step explanation:
30. One segment of the Mayan civilization started in 300 B.C.E. and ended in 250 C.E. How long did that segment of the Mayan civilization last?
A. 50 years
B. 300 years
C. 250 years
D. 550 years
Answers
Answer:
D. 550 years
Step-by-step explanation:
Firstly, you need to understand the meaning of B.C.E. and C.E. B.C.E. means "Before common era", where "common era" is the year where Jesus Christ was born (I will just call it "The Year" since the name is so long). The date has been used as a dating system for many years.
In this case, 300 B.C.E. means 300 years before "The Year" whereas 250 C.E. means 250 years after "The Year". Thus, the total would be:
300 + 250 = 550 years
Why is it so hard and not easy to understand people add so much new things every day
Answers
Answer:
your question makes no sense
Cecil sold vegetables at the farmers market on Saturday and Sunday. On Saturday, he collected twenty $1 bills, six $5 bills, four $10 bills, and five $20 bills. On Sunday, he collected thirty $1 bills, four $5 bills, five $10 bills, and one $20 bill. How much money did Cecil collect in total?
Answers
Answer:
$250.00
Step-by-step explanation:
twenty $1 bills- 20 x 1 = 20
six $5 bills- 6 x 5 = 30
four $10 bills- 4 x 10 = 40
five $20 bills- 5 x 20 = 100
thirty $1 bills- 30 x 1 = 30
four $5 bills- 4 x 5 = 20
five $10 bills- 5 x 10 = 50
one $20 bill- 1 x 20 = 20
Add the bold totals together and get the final answer of $250.00
Research has shown that competent communicators achieve effectiveness by
a. using the same types of behavior in a wide variety of situations.
b. developing large vocabularies.
c. apologizing when they offend others.
d. giving lots of feedback.
e. adjusting their behaviors to the person and situation.
Answers
Research has shown that competent communicators achieve effectiveness by adjusting their behaviors to the person and situation (option e).
Effective communication involves being adaptable and responsive to the specific context, individual preferences, and the needs of the situation.
Competent communicators recognize that different people have different communication styles, preferences, and expectations. They understand the importance of tailoring their communication approach to effectively connect and engage with others.
This may involve using appropriate language, tone, non-verbal cues, and listening actively to understand the needs and perspectives of others.
By adapting their behaviors, competent communicators can build rapport, foster understanding, and promote effective communication exchanges. They are mindful of the social and cultural dynamics at play, and they strive to communicate in a way that is respectful, inclusive, and conducive to achieving mutual goals.
In summary, competent communicators understand that effective communication is not a one-size-fits-all approach. They adjust their behaviors to the person and situation, demonstrating flexibility and adaptability in order to enhance communication effectiveness.
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Competent communicators achieve effectiveness mostly by adjusting their behaviors to suit the person they are communicating with and the situation they find themselves in. While other factors, like having a broad vocabulary or giving feedback, play a part in effective communication, the former is considered the most crucial.
Explanation:Research suggests that competent communicators achieve effectiveness mostly through adjusting their behaviors depending on the person they are communicating with and the situation they are in. This is option e. of your question. Communicating effectively involves behaviors like active listening, understanding the other person's point of view, being able to express thoughts and ideas clearly, and being polite and respectful. While a broad vocabulary (option b.) can be useful, it is not as crucial as adapting your behavior to fit the situation. Moreover, giving feedback (option d.) is a part of effective communication but not the sole defining factor. Apologizing when offending others (option c.) is also important but it doesn't necessarily make one a competent communicator. Using the same type of behavior in various situations (option a.) might not always work, as different situations and individuals require different communication styles.
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Poease help! Thank you
Answers
Answer:
28 and 12t
Step-by-step explanation:
4 x 7
4 x 3t
Answer:
28+12t
Step-by-step explanation:
Simplify the expression :)
btw you spelled please wrong
Which of these expressions is equivalent to 3(x−2) ?
Multiple choice question.
PLZS HELP I GIVE SOMEONE A BRAINLIEST
Answers
Step-by-step explanation:
3(x-6) is the correct answer to your question :)
Answer:
3x - 6 or 3x -6
Step-by-step explanation:
it is a distrubuve property so u multiply the numbers in the bracket by 3.so 3 time x is 3x.and the other one if it is -2 it is -6 but if its just 2 it is 6.
Hope it helped.
Simplify: -3x³ + 9x² + 30x -3x³-18x² - 24x ;x*-4, -2, 0
Answers
The simplified polynomial is -3x(x+2)(2x-1)
Given is a polynomial we need to simplify it,
-3x³ + 9x² + 30x -3x³-18x² - 24x
= -6x³ + 9x² + 30x -18x² - 24x
= -6x³-9x²+6x
= -3x(2x²-3x+2)
= -3x(x+2)(2x-1)
Hence the simplified polynomial is -3x(x+2)(2x-1)
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PLEASE HELPP!!
(look at the picture)
Answers
Answer:
Ok
Step-by-step explanation:
Answer: C. Taking a trip on a raft is safe if the weight of guests and their belongings (x pounds) and the weight of the guide and her equipment (250 pounds) together are less than 1,200 pounds
Step-by-step explanation:
Rewrite without parentheses. -3y^4(6y^3 -7y^2 - 3y)( click/tap on image to see.)
Answers
Answer:
I got -18y^7 + 21y^6 + 9y^5
Step-by-step explanation:
the top box has a length of 6 yards
Answers
Answer:
42 yd²
Step-by-step explanation:
Area of a rectangle = width x length
lateral surface area is the area of the sides only
So the difference between the total surface area and the lateral surface area would be the surface area of the ceiling and floor (top and bottom).
area of floor and ceiling = 2(3.5 x 6) = 42 yd²
Or, we can calculate the total surface area and lateral surface area, then find the difference:
total surface area = 2(3.5 x 4.2) + 2(4.2 x 6) + 2(3.5 x 6) = 121.8 yd²
lateral surface area = 2(3.5 x 4.2) + 2(4.2 x 6) = 79.8 yd²
difference = 121.8 - 79.8 = 42 yd²
Tony picked 5 1/2 bushels of tomatoes. He gave 1 3/8 bushels to his grandmother and sold the rest. Tony sold Answer bushels of tomatoes
Answers
Answer:
4 1/8
Step-by-step explanation:
In how many ways can the 6 students with an identical twins on a round table?
Answers
Answer:
120
Step-by-step explanation:
5*4*3*2*1
The summer and winter solstices are the longest day and night of the year, respectively. The summer solstice happens between June 20 and June 21, while the winter solstice is between December 21 and December 23. At a latitude of 28° the summer solstice is 14.5 hours long and the winter solstice is 14 hours long. Write a sinusoidal equation modeling the hours of daylight in a year. Use t for the variable and measure t in weeks. For the purposes of this problem, idealize a year to be 52 weeks long, with the winter solstice a week before the new year. h(t) = __________
Answers
The correct answer is h(t) = 14.25 + 0.25 * sin((π/26) * t)
To model the hours of daylight in a year using a sinusoidal equation, we can consider the average length of daylight, the amplitude of the variation, and the period of the function.
Given:
Summer solstice (longest day) is 14.5 hours
Winter solstice (longest night) is 14 hours
Let's analyze the information:
Average daylight: The average daylight throughout the year would be the average of the summer and winter solstices. It can be calculated as:
Average daylight = (14.5 + 14) / 2 = 14.25 hours
Amplitude: The difference between the average daylight and the longest day or night is the amplitude. In this case, the amplitude is half the difference between the summer and winter solstices. It can be calculated as:
Amplitude = (14.5 - 14) / 2 = 0.25 hours
Period: We are modeling the function over a year, which we'll consider as 52 weeks. Since we want to measure time in weeks (t), the period of the function is 52 weeks.
Putting all the information together, we can write the sinusoidal equation:
h(t) = Average daylight + Amplitude * sin((2π / Period) * t)
Substituting the values we calculated:
h(t) = 14.25 + 0.25 * sin((2π / 52) * t)
Simplifying further, the sinusoidal equation modeling the hours of daylight in a year is:
h(t) = 14.25 + 0.25 * sin(π/26 * t)
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in one statistics course students reported studying an average of 9.92 hours a week with a standard deviation of 4.54. treating this class as the population, what is the precentile for a student in the class who studes 8 hourse a week
Answers
The precentile for a student in the class who studies 8 hours a week is 5.74%.
To find the percentile for a student who studies 8 hours a week, we need to first standardize the student's study time by subtracting the mean study time and dividing it by the standard deviation of study time. This will give us the student's z-score. We can then look up the student's z-score in a standard normal table to find the percentile.
The student's z-score is (8 - 9.92) / 4.54 = -1.67.
Looking up -1.67 in a standard normal table, we find that the percentile is about 5.74%. This means that about 5.74% of the students in the class studied for less than 8 hours per week.
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As per the given standard deviation, the percentile for a student in the class who studies 8 hours a week is 50.80%
The term standard deviation in math is known as a measure which shows how much variation from the mean exists.
Here we have given that in one statistics course students reported studying an average of 9.92 hours a week with a standard deviation of 4.54.
Through this details, we have identified that the value of the following are given that,
=> weekly study time = 10 hours
=> average study time = 9.92 hours
=> standard deviation = 4.54 hours
Then the formula to calculate the Test statistic is written as
=> z = (weekly study time - average study time)/standard deviation
Apply the values on it, then we get
=> z = (10 - 9.92)/4.54
=> z = 0.08/4.54 = 0.02
While we take the value of z table, then we get the Cumulative area of the test statistic is 0.5080.
And the percentile is calculated by multiply the value by 100 as,
=> 0.5080 × 100 = 50.80
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