Answer:
Step-by-step explanation:
Given that Alice and Bob race two toy trains around a circular track. The trains move in the same direction and they meet every 120 seconds. If Alice's and Bob's toy trains move in opposite directions, at constant rates, they meet every 30 seconds
Circular track is 1800 m.
Let speed be x and y respectively
When in same direction relative speed = x-y and when in opposite directions =x+y
30(x+y) = 1800
x+y= 60
120(x-y) = 1800
x-y = 15
Solving x = 37.5 m/hr and y = 12.5 per hour.
When Alice and Bob's toy trains move in the same direction, their relative speed is the sum of their individual speeds. When they move in opposite directions, their relative speed is the difference between their individual speeds. By solving simultaneous equations, we find that Alice's toy train has a speed of 75 m/s and Bob's toy train has a speed of 60 m/s.
Explanation:Let's assume the speed of Alice's toy train is x m/s and the speed of Bob's toy train is y m/s.
When the trains move in the same direction, they meet every 120 seconds. Since they meet once every 30 seconds when moving in opposite directions, their relative speed is the sum of their individual speeds. So, when they move in the same direction, their relative speed will be x + y and when they move in opposite directions, their relative speed will be x - y.
When they move in the same direction, the distance covered by Alice's toy train in 120 seconds will be equal to the distance covered by Bob's toy train in 120 seconds, which is equal to the length of the circular track (1800 m). So, the equation can be written as:
x × 120 = y × 120 = 1800Simplifying the equation, we get:
x + y = 1800/120 = 15Similarly, when they move in opposite directions, the distance covered by Alice's toy train in 30 seconds will be equal to the distance covered by Bob's toy train in 30 seconds:
x × 30 = y × 30 = 1800Simplifying the equation, we get:
x - y = 1800/30 = 60Solving the equations simultaneously, we find that x = 75 m/s and y = -60 m/s. Since speed is always positive, we consider the magnitude of the velocity. Therefore, the speed of Alice's toy train is 75 m/s and the speed of Bob's toy train is 60 m/s.
What is not a problem modular division is used for?
Modular division, also known as modular arithmetic, has various applications in cryptography, computer science, and number theory. It is used in encryption algorithms like RSA and in solving congruence equations.
Explanation:Modular division, also known as modular arithmetic, is a mathematical operation that involves finding the remainder when one number is divided by another. Modular division is commonly used in various mathematical applications such as cryptography, computer science, and number theory.
One application of modular division is in the encryption and decryption of data using modular arithmetic. For example, the RSA encryption algorithm relies on modular division to encode and decode messages.
Another application of modular division is in solving congruence equations. Congruence equations represent the idea of equivalence of numbers modulo a given number. By using modular division, we can determine the solutions to these equations.
Learn more about Modular division here:https://brainly.com/question/31693612
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Modular division is used to find remainders in mathematics, but it is not used for dividing polynomials or dividing by zero. It's also not useful for dividing vectors in physics.
Modular division is a mathematical method used to find the remainder of a division problem, but it is not necessarily applicable in all contexts. For example, dividing a polynomial by a polynomial generally does not result in a polynomial, indicating that other methods must be used in this situation. Furthermore, it's important to note that modular division cannot be used to divide by zero, since this is undefined in mathematics. Additionally, within the discipline of physics, modular division is not applicable for operations such as dividing a vector by another vector component by component, as there are no useful physics applications for this type of operation.
Heron wants to buy a video game. The price is regularly priced at 55 dollars. The store has a discount of 20% off and a sales tax of 6%. How much will Heron pay for the video game
The amount paid by Heron for the video game is $46.64.
Step-by-step explanation:
Here, the marked price of the video game = $55
The discount percentage on the video game = 20%
Calculating 20% of the $55, we get:
[tex]\frac{20}{100} \times 55 = 11[/tex]
So, the discount offered on the video game = $11
Selling Price = Marked Price - Discount
=$55 - $11 = $44
Now, the tax percentage on the video game = 6%
Calculating 6% of the $44, we get:
[tex]\frac{6}{100} \times 44 =2.64[/tex]
So, the tax on the video game = $2.64
New Selling Price = Selling Price + Tax
=$44 + $2.64 = $46.64
So, the amount paid by Heron for the video game is $46.64.
Beth is writing out the steps using the "Shortest Route Algorithm". She just finished writing out all the routes for the third step. What route should she circle next?
Group of answer choices
AD; 8
ACE; 6
ACBE; 8
ACBD; 7
Answer:
ACBD; 7Explanation:
The "Shortest Route Algorigtm" aims to determine the most efficient or short route, when a several alternative pahtways can connect or be used to implement a solution.
A graph is drawn with the different nodes and paths that connect them. The distance between every pair of consecutive nodes is written.
The picture shows that for the step #1, there are, in principle, three routes: AB, AC, and AD.
AB must be discarded because it is not viable (a negative distance is not possible).
AC is more efficient than AD because the distance of AC is 3 and the distance of AD is 8. Thus AC is selected and circled.
To continue from AC, the possible routes are shown in step #2. They are ACB; 3 and ACE; 6.
ACB i s shorter, thus ACB is circled.
In step #3, the possible routes are ACBE; 8 and ACBD; 7. Thus, route ACBD is shorter, and it shall be circled.
The conclusion of the algorithm is that the route ACBD is the shoretes (most efficient).
The route to circle next is route ACBD; 7
From the question, we understand that she wants to determine the shortest route.
This means that, she has to circle the node with the smallest value in each step.
From the diagram, the smallest node in step 3 is ACBD; 7
Hence, the route to circle next is route ACBD; 7
Read more about algorithms at:
https://brainly.com/question/24793921
Is the product of 25 and 3/5 more or less than 14? Explain your answer. Write the product of 3/5 and 25.
Answer:
More. The product of 25 and 3/5 is 15 which is greater than 14.
Step-by-step explanation:
Need help on problem 40 part b for integrating in respect to y! Thanks!
Answer: [tex]\bold{(a)\quad \dfrac{32}{3}\qquad (b)\quad \dfrac{32}{3}}[/tex]
Step-by-step explanation:
(a) First, find the x-coordinates where the two equations cross
y = -1 and y = 3 - x²
-1 = 3 - x²
-4 = -x²
4 = x²
± 2 = x → These are the upper and lower limits of your integral
Then subtract the two equations and integrate with upper bound of x = 2 and lower bound of x = -2
[tex]\int_{-2}^{+2}[(3-x^2)-(-1)]dx\\\\\\=\int_{-2}^2(4-x^2)dx\\\\\\=4x-\dfrac{x^3}{3}\bigg|_{-2}^{+2}\\\\\\=\bigg(8-\dfrac{8}{3}\bigg)-\bigg(-8+\dfrac{8}{3}\bigg)\\\\\\=\large\boxed{\dfrac{32}{3}}[/tex]
(b) We know the upper and lower bounds of the y-axis as y = 3 and y = -1
Next, find the equation that we need to integrate by solving for x.
y = 3 - x²
x² + y = 3
x² = 3 - y
x [tex]=\pm\sqrt{3-y}\\[/tex]
[tex]\rightarrow \qquad x=\sqrt{3-y}\quad and \quad x=-\sqrt{3-y}[/tex]
Now, subtract the two equations and integrate with upper bound of y = 3 and lower bound of y = -1
[tex]\int_{-1}^{+3}[(\sqrt{3-y})-(-\sqrt{3-y})]dy\\\\\\=\int_{-1}^{+3}(2\sqrt{3-y})dy\\\\\\=\dfrac{-4\sqrt{(3-y)^3}}{3}\bigg|_{-1}^{+3}\\\\\\=\bigg(0\bigg)-\bigg(-\dfrac{32}{3}\bigg)\\\\\\=\large\boxed{\dfrac{32}{3}}[/tex]
A telemarketer calls people and tries to sell them a subscription to a daily newspaper. On 22% of her calls, there is no answer or the line is busy. She sells subscriptions to 10% of the remaining calls. For what proportion of calls does she make a sale? Give your answer as a decimal, and do not round.
Answer:
She makes 7.8% portion of sales collectively.
Step-by-step explanation:
Let us assume she makes total of 100 calls.
The percentage of calls which are busy or unanswered = 22 %
So, now calculating 22% of 100 , we get:
[tex]\frac{22}{100} \times 100 = 22[/tex]
So, 22 of her calls are UNANSWERED.
Now, let us find out the number of calls successfully made by telemarketer.
Total successful calls = Total Calls made - Number of unanswered calls
= 100 - 22 = 78
So, she makes a total of 78 calls successfully.
Now, he sells subscriptions to 10% of the 78 calls made successfully.
So, now calculating 10% of 78 , we get:
[tex]\frac{10}{100} \times 78 = 7.8[/tex]
So, she sold 7.8 subscriptions in total out of 100 attempts.
Also, as we know 7.8 out of 100 = 7.8% of 100.
Hence, she makes 7.8% portion of sales collectively.
Juan purchased an antique that had a value of \$200$200dollar sign, 200 at the time of purchase. Each year, the value of the antique is estimated to increase 10\, percent over its value the previous year. The estimated value of the antique, in dollars, 222 years after purchase can be represented by the expression 200a200a200, a, where aaa is a constant. What is the value of aaa?
Final answer:
The antique purchased by Juan increases in value by 10% each year. The value of the antique after 2 years can be found by calculating the expression $200a, where a is a constant. The value of [tex]\( a \) is \( 1.21 \).[/tex]
Explanation:
To find the value of a, we need to represent the annual increase of 10% in terms of multiplication.
After the first year, the value of the antique increases by [tex]\( 10\% \) of its previous value, which is \( 0.10 \times 200 \) dollars.[/tex]
After the second year, the value of the antique increases by [tex]\( 10\% \) of its value at the end of the first year, which is \( 0.10 \times (200 + 0.10 \times 200) \) dollars.[/tex]
Generally, after \( n \) years, the value of the antique will be [tex]\( 200 \times (1 + 0.10)^n \) dollars.[/tex]
The expression given for the value of the antique 2 years after purchase is 200a , where a is a constant. This represents the value of the antique after 2 years.
Equating the expression to the value of the antique after 2 years, we have:
[tex]\[ 200a = 200 \times (1 + 0.10)^2 \][/tex]
Now, let's solve for \( a \):
[tex]\[ 200a = 200 \times (1.10)^2 \][/tex]
200a = 200 \times 1.21
200a = 242
Dividing both sides by 200:
[tex]\[ a = \frac{242}{200} \][/tex]
[tex]\[ a = 1.21 \][/tex]
Therefore, the value of [tex]\( a \) is \( 1.21 \).[/tex]
A recycling bin is in the shape of a right rectangular prism. The bin is 12 meters long, 5 1/2 meters wide, and 6 1/2 meters tall. What is the volume of the recycling bin? Omg Help me!Please i dont get this?
Answer: The volume is 143
Step-by-step explanation:
Of the two production methods, a company wants to identify the method with the smaller population mean completion time. One sample of workers is selected and each worker first uses one method and then uses the other method. The sampling procedure being used to collect completion time data is based on
a.
matched samples.
b.
worker samples.
c.
pooled samples.
d.
independent samples.
Answer:
a) matched samples.
Step-by-step explanation:
Matched samples (also known as matched pairs, paired samples or dependent samples) are those samples which can be matched in pairs for one set of item and the sample data are not independent of each other. The pairs don’t have to be different people, it could be the same individuals at different time or tested on different activities for example
sampling the blood pressures of the same people before and after they receive a dosethe same people being measured when the group is given two different tests at different timesThe completion time data collection method used by the company, which involves each worker using both production methods, is based on matched samples.
Explanation:The current method used to collect completion time data involves each worker being required to use both production methods. This method is known as matched samples. It involves using the same subjects in two different conditions to measure the difference in outcomes. In this case, the matched samples design is being used to compare the completion times of the workers using both production methods. The design is a test of dependent means, classified as a matched pairs design. This design is useful in situations where the same subject is being tested in two different conditions. The matched pairs design allows for a more accurate comparison of the two conditions, as it eliminates the variability between different subjects.
PLS HELP What is the value of cos (sin−1(−0.435))?
Answer:
0.900
Step-by-step explanation:
The easiest way to solve this problem is by putting it into a calculator.
When put into a calculator, the answer comes out to be 0.9004304526, which can be rounded to 0.900, or just 0.9.
Good morning ☕️
Answer:
0.900
Step-by-step explanation:
using a calculator you’ll find:
sin⁻¹(-0.435) = -25.785293878311
now
cos(sin⁻¹(-0.435)) = cos(-25.785293878311)
= 0.900430452617
If we round 0.900430452617 to nearest thousandth we get: 0.900
:)
find the coordinate of U' after a 90° counterclockwise rotation of the triangle about the origin and then a translation of 2 units down and 5 units left.
Answer: U' = (0, 1)
Step-by-step explanation:
U = (3, -5)
rotate 90° counterclockwise means (x, y) = (-y, x)
new U = (5, 3)
down 2 units means subtract 2 from the y-coordinate
newer U = (5, 1)
left 5 units means subtract 5 from the x-coordinate
U' = (0, 1)
Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates. If Machine A’s speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates. How many hours does it currently take Machine A to produce 1 widget on its own?
[tex]\boldsymbol{\mathbf{Answer}}[/tex]
[tex]\boldsymbol{\mathbf{Machine \, A \,will\, take \,6 \,hours\, to \,produce\, 1 \,widget \,on\, its\, own.}}[/tex]
[tex]\boldsymbol{\mathbf{Step-by-step \,explanation:}}[/tex]
Let,
performance rate of machine A is x widget per hour.
performance rate of machine A is y widget per hour.
As given, Machine A and Machine B can produce 1 widget in 3 hours working together.
I.e mathemetically,
[tex]\boldsymbol{x + y=\frac{1}{3}......(1)}[/tex]
lly for second statement, Machine A’s speed were doubled, the two machines could produce 1 widget in 2 hours working together.
i.e mathematically,
[tex]\boldsymbol{2x + y=\frac{1}{2}......(2)}[/tex]
Substact equation (1) in (2)
[tex]x + y=\frac{1}{3}[/tex]
[tex]-2x + y=\frac{1}{2}[/tex]
Resultant equation will be,
[tex]-x=\frac{-1}{6}[/tex]
[tex]\boldsymbol{x = \frac{1}{6}}[/tex]
Performance rate of machine A is \frac{1}{6} widget per hour.
what is time Machine A will take to produce 1 widget on its own.
i.e = [tex]\frac{1}{\frac{1}{6}}[/tex]
[tex]\boldsymbol\mathbf{{=\, 6 \,hours.}}[/tex]
Jolene drove to a state park. She drove 1/4 of the distance the first day. She drove farther the second day. What pat of the distance might Jolene have driven the second day?
Answer: [tex]\frac{2}{4}[/tex] or [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
You need to analize the information given in the exercise.
Let be "x" represents the whole distance part that Jolene have driven to the state park.
According the the explained in the problem, in the first day Jolene drove [tex]\frac{1}{4}[/tex] of the distance.
Knowing that, you can identify that the whole distance (or the value of "x"), is the following:
[tex]x=\frac{4}{4}[/tex]
(If you simplify it, you get: [tex]x=1[/tex])
You also know that the second day Jolene drove farther than the first day; therefore, there are two possible cases for the part of the distance she might have driven the second day. These cases are:
Case 1: [tex]\frac{2}{4}[/tex] of the distance the second day.
Case 2: [tex]\frac{3}{4}[/tex] of the distance the second day.
Which scatterplot has a negative r value? There are 3 graphs
Answer:
Step-by-step explanation:
The relationship is negative, negative correlation
The data shown represent the number of runs made each year during Bill Mazeroski’s career. Check for normality.
30 59 69 50 58 71 55 43 3
66 52 56 62 36 13 29 17 31
Answer:
The given data is not normal.
Step-by-step explanation:
We are given the following data:
30, 59, 69, 50, 58, 71, 55, 43, 3, 66, 52, 56, 62, 36, 13, 29, 17, 31
Condition for normality:
Mean = Mode = Median
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{800}{18} = 44.44[/tex]
Mode is the most frequent observation of the data.
Since all the value appeared once, there is no mode.
[tex]Median:\\\text{If n is odd, then}\\\\Median = \displaystyle\frac{n+1}{2}th ~term \\\\\text{If n is even, then}\\\\Median = \displaystyle\frac{\frac{n}{2}th~term + (\frac{n}{2}+1)th~term}{2}[/tex]
Sorted data: 3, 13, 17, 29, 30, 31, 36, 43, 50, 52, 55, 56, 58, 59, 62, 66, 69, 71
Median =
[tex]=\dfrac{9^{th}+10^{th}}{2} = \dfrac{50+52}{2}=51[/tex]
Since the mean, mode and median of data are not equal, the data is not normal.
). Bees are one of the fastest insects on Earth. They can fly 22 miles in 2 hours, and 55 miles in 5 hours. Write an algebraic expression to show how many miles a bee can fly in h hours. If a bee flies 4 hours at this speed, how many miles will it travel?
Answer:
Step-by-step explanation:
Look at the info given as (x, y) coordinates with the number of hours as x and the number of miles as y. The first coordinate then is (2, 22) and the second is (5, 55). The rate at which the bee flies is the same as the slope of the coordinates.
[tex]\frac{55-22}{5-2}=11[/tex]
This means that the bee flies 11 miles per hour. Use either one of the coordinates now to find the equation for the line. I pick (2, 22) and point-slope form:
y - 22 = 11(x - 2) and
y - 22 = 11x - 22 so
y = 11x
That's the equation. If we want to use it as a model, we can find how many miles it will fly in a given time, or how long it will take to fly a given number of miles. We are asked to find how far it can fly in 4 hours. So we will use our equation and replace x with 4:
y = 11(4) so
y = 44 miles
I am confused about the wording on this problem and also when I used the Pythagorean theorem it came out as wrong.
Answer:
Step-by-step explanation: they are asking “What is x + 3 + y”. So use Pythagorean’s theorem to get x (it should be 4) and then find y and I think u get square root of 13, then add 4 + 3 + square root 13
The population of a city is expected to increase by 7.5% next year. If p represents the current popultion, which expression represents the expected populations next year?
Answer: P = Po ( 1 + 0.075)
Step-by-step explanation: let Po = initial population
P = final population.
The increase in population is by 7.5%, which implies that if the initial population Increases by 7.5%, we would have a new (current) population.
Final population = initial population + increament of initial population.
Where increment of initial population = 7.5% of Po = 0.075 Po
P = Po + 0.075Po
P = Po ( 1 + 0.075)
Using the distance formula, d = √(x2 - x1)2 + (y2 - y1)2, what is the distance between point (-2, 2) and point (4, 4) rounded to the nearest tenth?
5.7 units
4 units
6.3 units
1 unit
Answer:
The distance is 6.3 units
I really need help oof-
Angle α lies in quadrant II, and tanα=−12/5 . Angle β lies in quadrant IV, and cosβ=3/5.
What is the exact value of cos(α−β) ?
Enter your answer in the box.
cos(α−β) = __
From the given info (and the linked question) we find
[tex]\cos\alpha=-\dfrac5{13}[/tex]
[tex]\sin\alpha=\dfrac{12}{13}[/tex]
[tex]\sin\beta=-\dfrac45[/tex]
Then using the angle-sum identity for cosine, we have
[tex]\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta[/tex]
[tex]\cos(\alpha-\beta)=\left(-\dfrac5{13}\right)\dfrac35+\dfrac{12}{13}\left(-\dfrac45\right)=-\dfrac{63}{65}[/tex]
Final answer:
To find the exact value of cos(α-β), we use the cosine sum and difference identity and the respective sine and cosine values calculated from the given tangent and cosine values for angles α and β. Using this approach, we find that cos(α-β) equals 63/65.
Explanation:
The exact value of cos(α-β) can be found by using the sum and difference identities for cosine. Since tanα = -12/5, in quadrant II, we can find the corresponding sine and cosine values for α using the Pythagorean identity sin²α + cos²α = 1. For cosβ = 3/5, in quadrant IV, we do a similar procedure to find the sine of β. With both sine and cosine for α and β, we use the identity cos(A-B) = cosA cosB + sinA sinB to find cos(α-β).
To find the sine and cosine for α, given that tanα = -12/5, we know that the opposite side is -12, and the adjacent side is 5, so the hypotenuse using the Pythagorean theorem is √(12² + 5²) = √(144+25) = √169 = 13. Thus sinα = -12/13 (negative because α lies in the second quadrant where sine is negative) and cosα = 5/13 (positive because cosine in the second quadrant is positive).
For β, we already have cosβ = 3/5. The sine can be found using the Pythagorean identity 1 - cos²β = sin²β, which gives sinβ = -√(1 - (3/5)�) = -√(1 - 9/25) = -√(16/25) = -4/5 (negative because β is in the fourth quadrant where sine is negative).
Now, we can find the exact value of cos(α-β) by plugging in the values:
cos(α-β) = cosα cosβ + sinα sinβ = (5/13)*(3/5) + (-12/13)*(-4/5)
= 15/65 + 48/65 = 63/65.
Therefore, the exact value of cos(α-β) is 63/65.
An object is traveling at a steady speed of 10 and one tenth km/h. How long will it take the object to travel 4 and nine tenths km ? First round to the nearest integer to find the estimated answer. Then find the exact answer.
Final answer:
To find the time it will take for an object to travel a certain distance at a given speed, divide the distance by the speed. The estimated time to travel 4.9 km at a speed of 10 km/h is approximately 0.5 hours. The exact time, considering the speed as 10.1 km/h, is also approximately 0.5 hours.
Explanation:
To find the time it will take for an object to travel a certain distance at a given speed, we can use the formula:
Time (in hours) = Distance (in kilometers) / Speed (in kilometers per hour)
First, let's round the speed to the nearest integer, which is 10 km/h. To estimate the time it will take to travel 4.9 km, we can divide the distance by the estimated speed:
Estimated Time = 4.9 km / 10 km/h ≈ 0.49 hours ≈ 0.5 hours
To find the exact time, we will use the given speed of 10 and one-tenth km/h. We can convert this speed to decimal form, which is 10.1 km/h. Now, we can calculate the exact time:
Exact Time = 4.9 km / 10.1 km/h ≈ 0.485 hours ≈ 0.5 hours
Therefore, it will take approximately 0.5 hours or 30 minutes for the object to travel 4.9 km.
In the past, every ten-percentage-point increase in cigarette prices in the country of Coponia has decreased per capita sales of cigarettes by four percent. Coponia is about to raise taxes on cigarettes by 9 cents per pack. The average price of cigarettes in Coponia is and has been for more than a year 90 cents per pack. So the tax hike stands an excellent chance of reducing per capita sales of cigarettes by four percent.
Which of the following is an assumption on which the argument depends?
A. Tobacco companies are unlikely to reduce their profit per pack of cigarettes to avoid an increase in the cost per pack to consumers in Coponia.
B. Previous increases in cigarette prices in Coponia have generally been due to increases in taxes on cigarettes.
C. Any decrease in per capita sales of cigarettes in Coponia will result mainly from an increase in the number of people who quit smoking entirely.
D. At present, the price of a pack of cigarettes in Coponia includes taxes that amount to less than ten percent of the total selling price.
E. The number of people in Coponia who smoke cigarettes has remained relatively constant for the past several years.
Answer:
The assumption will depend on the argument that C. Any decrease in per capita sales of cigarettes in Coponia will result mainly from an increase in the number of people who quit smoking entirely.
Step-by-step explanation:
Per capita income or average income measures the average income earned per person in a given area in a specified year. It is calculated by dividing the area's total income by its total population. Per capita income is national income divided by population size.
Tax is a compulsory contribution to state revenue, levied by the government on workers' income and business profits, or added to the cost of some goods, services, and transactions.
Two poles are connected by a wire that is also connected to the ground. The first pole is 20 ft tall and the second pole is 10 ft tall. There is a distance of 30 ft between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire need
Answer:
Therefore the wire should be anchored at 10 ft away from pole which is 10 ft long.
Step-by-step explanation:
Given that , The distance between two poles is 30 ft.
The length of 1st pole is = 20 ft
The length of second pole is = 10 ft.
Let the wire anchored to the ground at a distance x ft from the second pole.
Then, the distance of anchored from the first pole is = (30-x)
The total length of the wire is L = m+n
We know the pythagorean theorem,
Height²+base² = hypotenuse²
To find the value of m and n we use pythagorean theorem
From the left side triangle in the picture we get,
10²+x²= m²
⇒m²=100+x²
[tex]\Rightarrow m= \sqrt {100+x^2[/tex]
and right side triangle in the picture we get,
20²+(30-x)² = n²
⇒n²= x²-60x+1300
[tex]\Rightarrow n= \sqrt {x^2 -60x+1300}[/tex]
Then ,
[tex]L= \sqrt{(100+x^2)}+\sqrt{(x^2-60x+1300) }[/tex]
Differentiating with respect to x
[tex]L'= \frac {2x}{2\sqrt{100+x^2}}+ \frac{2x-60}{2\sqrt {x^2-60x+1300}}[/tex]
For minimize, L' =0
[tex]\frac {2x}{2\sqrt{100+x^2}}+ \frac{2x-60}{2\sqrt {x^2-60x+1300}}=0[/tex]
[tex]\Rightarrow \frac {x}{\sqrt{100+x^2}}=- \frac{x-30}{\sqrt {x^2-60x+1300}}[/tex]
Squaring both sides
[tex]\Rightarrow( \frac {x}{\sqrt{100+x^2}})^2=(- \frac{x-30}{\sqrt {x^2-60x+1300}})^2[/tex]
[tex]\Rightarrow x^2(x^2-60x+1300)= (x^2-60x+900)(100+x^2)[/tex]
[tex]\Rightarrow x^4 -60x^3+1300x^2= 100x^2-6000x+90000+x^4-60x^3+900x^2[/tex]
[tex]\Rightarrow 300x^2+6000x-90000=0[/tex]
[tex]\Rightarrow x^2+20x-300=0[/tex]
[tex]\Rightarrow x=10,-30[/tex]
Therefore x = 10. [x=-30 negligible, since distance can not negative]
Therefore the wire should be anchored at 10 ft away from pole which is 10 ft long.
The problem can be solved geometrically through the principles of trigonometry. By setting up two right triangles formed by the telephone poles and the anchoring point, we can create two equations by Pythagorean Theorem. By taking the derivative of the total wire length and setting it to zero, we can find the optimal value for 'x' (location of the anchoring point) which results in the minimal amount of wire used.
Explanation:To solve for the minimal amount of wire needed, we can use the principles of mathematics. More specifically, we will use the concept of trigonometry and geometry to create two right triangles. The taller pole (20ft), the shorter pole (10ft) and the point on the ground where the wire is anchored form the two right triangles, one with 20ft height and another with 10ft height.
Let's denote the length of wire between the taller pole and ground as 'a', between the shorter pole and the ground as 'b', and the distance between the point on the ground where the wire is anchored and the base of the first pole as 'x'. We have:
Relationship 1: a = sqrt((20)^2 + x^2), based on the Pythagorean theorem; Relationship 2: b = sqrt((10)^2 + (30 - x)^2)
The total length of wire used (which we want to minimize) is a + b.
To find the minimal length, we can take the derivative of 'a+b' with respect to 'x' and set the derivative equation to 0 then solve for 'x'. This will give you where to place the anchor on the ground (minimal amount of wire used) between the two poles. You may find out an optimal 'x' value that is less than 30ft, ensuring that the anchoring point is between the two poles.
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Option D is cut off but option d is 6a+16
Please Help
Answer:
A
Step-by-step explanation:
8(2) = 16
17 (2) = 34
34^2 = 16^2 + x^2
1156 = 256 + x^2
1156- 256 = x^2
900 = x^2
square root of 900 = x
x = 30
15(2) = 30
Answer:
A 15a
Step-by-step explanation:
This is a right triangle so we can use the Pythagorean theorem
a^2 +b^2 = c^2 where a and b are the legs and c is the hypotenuse
Letting the unknown side be x
(8a)^2 + x^2 = (17a)^2
64a^2 + x^2= 289a^2
Subtracting 64a^2 from each side
64a^2 -64a^2 + x^2= 289a^2-64a^2
x^2 =225a^2
Taking the square root of each side
sqrt(x^2) =sqrt(225a^2)
x = 15a
Mary collected $2.75 in nickels and dimes. Ten less than twice the number of nickels represents the number of dimes she has. How many of each kind of coin does she have?
Answer: There are 15 nickles and 20 dimes.
Step-by-step explanation:
Since we have given that
Let the number of nickles be 'x'
Let the number of dimes be '2x-10'.
Total amount = $2.75
According to question, we get that
[tex]0.1(2x-10)+0.05x=2.75\\\\0.2x-1+0.05x=2.75\\\\0.25x=2.75+1\\\\0.25x=3.75\\\\x=\dfrac{3.75}{0.25}\\\\x=15[/tex]
Hence, there are 15 nickles and [tex]2x-10=2(15)-10=30-10=20[/tex] dimes.
Therefore, there are 15 nickles and 20 dimes.
Final answer:
Mary has 15 nickels and 20 dimes.
Explanation:
Let's solve this problem step-by-step:
Let's assume the number of nickels as N and the number of dimes as D.According to the given information, the value of the nickels is 0.05N and the value of the dimes is 0.10D.The total amount collected is $2.75, so we can write the equation: 0.05N + 0.10D = 2.75According to the second piece of information, ten less than twice the number of nickels represents the number of dimes. So, we can write another equation: D = 2N - 10We now have a system of two equations with two variables:0.05N + 0.10D = 2.75D = 2N - 10Solve the system of equations to find the values of N and D.Substitute the value of D from the second equation into the first equation: 0.05N + 0.10(2N - 10) = 2.75Expand and simplify the equation: 0.05N + 0.20N - 1 = 2.750.25N - 1 = 2.750.25N = 3.75N = 3.75/0.25N = 15Substitute the value of N into the second equation to find the value of D: D = 2(15) - 10D = 30 - 10D = 20Therefore, Mary has 15 nickels and 20 dimes.:_____ requires constructing and applying statistical models that predict labor demand for the next year, given relatively objective statistics from the previous year. Select one: a. Propensity analysis b. A leading indicator c. A yield ratio d. Transitional matrix e. Trend analysis
Answer:
e) Trend Analysis
The term is 'trend analysis'. This refers to a statistical method used to evaluate and predict future trends based on historical data. This technique is specifically referenced in prediction of labor demand using prior year’s data.
Explanation:The concept referred to in the question is e. Trend analysis. Trend analysis is a statistical method used to evaluate and predict future trends based on historical data. In the context of labor demand, a trend analysis would involve examining labor demand data from the previous year, identifying patterns and trends within that data, and using statistical models to make predictions about labor demand for the upcoming year.
For example, suppose a company has seen a steady increase in labor demand over the past five years. Based on this trend, they can build a statistical model that predicts an increase in labor demand for the next year as well. This strategy helps companies plan for their future staffing needs, ensuring they have the necessary resources to meet their objectives.
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Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle A?
Enter your answer in the box.
°
Answer: [tex]m\angle A=116\°[/tex]
Step-by-step explanation:
The missing figure is attached.
For this exercise it is important to remember that, by definition, the opposite interior angles of an inscribed quadrilateral are supplementary, which means that their sum is 180 degrees.
Based on this, you can identify that the angle D and the angle B are opposite and, therefore, supplementary.
Knowing that, you can write the following equation:
[tex]x+28\°=180\°[/tex]
Now you must solve for "x" in order to find its value. This is:
[tex]x=180\°-28\°\\\\x=152\°[/tex]
Then:
[tex]m\angle D=152\°[/tex]
You know that:
[tex]m\angle A=(x-36)\°[/tex]
Therefore, since you know the value of "x", you can substitute it into [tex]m\angle A=(x-36)\°[/tex] and then you must evaluate, in order to find the measure of the angle A. This is:
[tex]m\angle A=152\°-36\°\\\\m\angle A=116\°[/tex]
A tank holds 50 gal of water, which drains from a leak at the bottom, causing the tank to empty in 20 min. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after t minutes as V(t)=50(1−t20)20≤t≤20 (a) Find V(0) and V(20). (b) What do your answers to part (a) represent? (c) Make a table of values of V(t) for t = 0, 5,10, 15, 20. (d) Find the net change in the volume V as t changes from 0 min to 20 min.
Answer:
(a) V(0) = 50 gal, V(20) = 0 gal
(b)At t= 0 the tank is full.
At t=0 the tank is empty
(c)
Time volume
0 50 gal
5 37.5 gal
10 25 gal
15 12.5 gal
20 0 gal
(d)
Net change of volume = 50 gal
Step-by-step explanation:
Given that the capacity of the tank is 50 gal.
Torricelli's Law gives the volume of water remaining in the tank after t minutes as
[tex]V(t)=50(1-\frac{t}{20})^2[/tex]
(a)
To find V(0), we put t = 0 in the above equation
[tex]V(0)=50(1-\frac{0}{20})^2[/tex]
[tex]=50(1-0)^2[/tex]
= 50 gal
To find V(20), we put t =2 0 in the above equation
[tex]V(20)=50(1-\frac{20}{20})^2[/tex]
[tex]=50(1-1)^2[/tex]
= 0 gal
(b)
At t= 0 the tank is full.
At t=0 the tank is empty.
(c)
Time V(t)
0 [tex]50(1-\frac{0}{20})^2=50 \ gal[/tex]
5 [tex]50(1-\frac{5}{20})^2=37.5 \ gal[/tex]
10 [tex]50(1-\frac{10}{20})^2=25 \ gal[/tex]
15 [tex]50(1-\frac{15}{20})^2=12.5 \ gal[/tex]
20 [tex]50(1-\frac{20}{20})^2=0[/tex]
(d)
Net change of volume = V(0) -V(20)
=(50-0) gal
= 50 gal
The volume V(t) of water remaining in the tank after t minutes is given by V(t) = 50(1−t/20). V(0) represents the initial volume of water in the tank, which is 50 gallons. V(20) represents the volume of water remaining in the tank after 20 minutes, which is 0 gallons.
Explanation:(a) To find V(0), substitute t = 0 into the equation V(t) = 50(1−t/20).
V(0) = 50(1−0/20) = 50(1−0) = 50(1) = 50
Similarly, to find V(20), substitute t = 20 into the equation V(t) = 50(1−t/20).
V(20) = 50(1−20/20) = 50(1−1) = 50(0) = 0
(b) V(0) represents the initial volume of water in the tank, which is 50 gallons. V(20) represents the volume of water remaining in the tank after 20 minutes, which is 0 gallons.
(c) Creating a table of values of V(t) for t = 0, 5, 10, 15, 20:
t | V(t)
--------------------
0 | 50
5 | 37.5
10 | 25
15 | 12.5
20 | 0
(d) The net change in volume V as t changes from 0 min to 20 min is V(20) - V(0).
V(20) - V(0) = 0 - 50 = -50 gallons
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Sanjay bought 12 granola bars,which was 4 times as many granola bars as Lena bought.which equation shows the number of granola bars,b,that Lena bought?
Evan has $0.45 worth of pennies and nickels. He has a total of 21 pennies and nickels altogether. Determine the number of pennies and the number of nickels that Evan has.
The number of pennies and nickels that has a worth of $0.45 is 15 and 6 respectively
Given:
total worth = $0.45
Total coins = 21
let
number of pennies = x
number of nickels = y
x + y = 21 (1)
0.01x + 0.05y = 0.45 (2)
multiply (1) by 0.01
0.01x + 0.01y = 0.21 (3)
0.01x + 0.05y = 0.45 (2)
subtract (2) from (1)
0.05y - 0.01y = 0.45 - 0.21
0.04y = 0.24
y = 0.24 / 0.04
y = 6
substitute y = 6 into (1)
x + y = 21 (1)
x + 6 = 21
x = 21 - 6
x = 15
Therefore, the number of pennies and nickels that has a worth of $0.45 is 15 and 6 respectively.
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