The velocities of particles are given by the derivatives of their displacement functions. Equating the velocity functions of the two particles and solving numerically, we find that they have the same velocity for the first time at about t = 1.569 seconds.
Explanation:The velocity of an object is given by the derivative of its displacement function. So, we need to first find the derivatives of the given displacement functions to find the velocities of the particles.
The derivative of e^(4cos(t)) is [tex]-4e^{(4cos(t))sin(t)[/tex]. The derivative of [tex]-(t^3)/(3) - (t^2)/(2) + 2 is -t^2 - t.[/tex] Now, we equate the two velocities and solve for t[tex].-4e^(4cos(t))sin(t) = -t^2 - t[/tex]
Unfortunately, this equation does not have a simple algebraic solution. However, it can be solved numerically using, for example, a graphing calculator or numerical software. By using these tools, we find the first positive time at which the particles have approximately the same velocity to be about t = 1.569 seconds. Therefore, the correct answer is A.) t = 1.569 seconds.
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x-y=2 and x+y=-2 sove by graphing please and show the solution
Answer:
The answer to your question is There is only one solution (0, -2)
Step-by-step explanation:
Data
Equation 1 x - y = 2
Equation 2 x + y = -2
Solve for y
Equation 1 y = x -2
y = x - 2
Equation 2 y = - x - 2
See the graph below
These lines cross in point (0 ,-2), that is the only solution.
If the lines have not crossed, they were parallel lines.
A certain college team has on its roster three centers, five guards, three forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created? [Hint: Consider lineups without X, then lineups with X as guard, then lineups with X as forward.]
Final answer:
Explaining the calculation of different starting lineups with and without a versatile player X on a college team roster.
Explanation:
The total number of different starting lineups can be created by considering various scenarios:
Without X: 3 centers, 5 guards, and 3 forwards = 3*5*3 = 45 lineupsX as a guard: 3 centers, 6 guards, and 3 forwards = 3*6*3 = 54 lineupsX as a forward: 3 centers, 5 guards, and 4 forwards = 3*5*4 = 60 lineupsTo get the final count, add up the lineups from each scenario: 45 (without X) + 54 (X as guard) + 60 (X as forward) = 159 different starting lineups.
Therefore, as per the above explaination, the correct answer is 159 different lineups
Find the present value. Assume there are 360 days in a year.
FV = $83870
t = 226 days
r = 6.8%
Step-by-step explanation:
[tex]PV = FV \div (1+i)^n[/tex]
Here FV = $83870
i = 6.8% = 0.068
[tex]n = \frac{226}{365}[/tex] =0.62
Therefore,
[tex]PV=\frac{83870}{(1+0.068)^{0.62}}[/tex]
=$80517.91
Therefore the present value = $ 80517.91
Geraldine is asked to explain the limits on the range of an exponential equation using the function f(x) = 2x. She makes these two statements: 1. As x increases infinitely, the y-values are continually doubled for each single increase in x. 2. As x decreases infinitely, the y-values are continually halved for each single decrease in x. She concludes that there are no limits within the set of real numbers on the range of this exponential function. Which best explains the accuracy of Geraldine’s statements and her conclusion? a.Statement 1 is incorrect because the y-values are increased by 2, not doubled. b.Statement 2 is incorrect because the y-values are doubled, not halved. The conclusion is incorrect because the range is limited to the set of integers. The conclusion is incorrect because
The true statement is: d. The conclusion is incorrect because the range is limited to the set of positive real numbers.
The function is given as:
[tex]\mathbf{f(x) =2x}[/tex]
The above function implies that:
When x increases by 1, y increases by 2When x decreases by 1, y decreases by 2The above highlights mean that: Geraldine's claims are incorrect.
Because y increases or decreases by 2, when x increases or decreases by 1
In other words, the value of y does not get doubled or halved.
Hence, both statements are incorrect
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The formula for the perimeter of a rectangle with length and width is as follows. Suppose the length of the rectangle is 5 times the width. Rewrite in terms of only. It is not necessary to simplify?
Answer:
P = 2(5W + W)
Step-by-step explanation:
P = 2(L +W)
L = 5W
P = 2(5W + W)
P = 10W + 2W
P = 12W
A pharmacist has a 6% solution of cough syrup and a 14% solution of the same cough syrup. How many ounces of each must be mixed to make 16 ounces of a 10% solution of cough syrup?
Answer:
8 ounces of each must be mixed to make 16 ounces of a 10%solution.
Step-by-step explanation:
The careful analysis and detailed calculation is as shown in the attached file.
Please help me!!!! Please show work!!!
Answer:
B=80 A=40 EFD=60 BCF=120
Step-by-step explanation:
First off chill. Second off B=80 A=40 EFD=60 BCF=120. It is quite simple. The congruency marks give the first angle away and you solve from there. If you need help I would do more research because this is important for later units. Also the triangles are congruent.
Answer
Angle A = 40 degrees
Angle B = 80 degrees
Measure BCF = 120 degrees
Measure EFD = 60 degrees
Step-by-step explanation:
Angle A:
Ok, so we know angle A is congruent to angle D because of the line. If you make an equation out of it, you get 2x+20 = 3x+10. If you solve this equation:
2x + 20 = 3x + 10
20 = x+10
10 = x
you will get x=10. Plug it into the equation, and Angle A = 40 degrees.
Angle B:
The two lines that are both on angle B and E mean that they are congruent. We know that angle E = 80 degrees, so angle B does too.
Measure BCF:
We know that Angle A = 40 degrees, and angle B = 80 degrees. The degree sum of all angles in a triangle is 180 degrees.
80 + 40 = 120
180 - 120 = 60
So measure BCA = 60 degrees.
That angle is on a straight line. Two angles on a straight line add up to 180 degrees.
180 - 60 = 120.
So, Measure BCF = 120 degrees.
Measure EFD:
We already found measure BCA while finding measure BCF, and that is just congruent to EFD.
So, measure EFD = 60 degrees
Frank,an nfl running back rushed for an average of 110 yards per game last season. This season, his average 40% higher. What is his average this season?
Answer:
His average this season is 154 yards a game.
Step-by-step explanation:
This question can be solved by a simple rule of three.
Last season's numbers(110 yards a game) is 100% = 1 decimal
This season number(x yards a game) is an increase of 40% over last season, so 40% + 100% = 140% = 1.40 decimal.
So
110 yards a game - 1
x yards a game - 1.40
[tex]x = 110*1.40 = 154[/tex]
His average this season is 154 yards a game.
If 4 fair 6-sided dice are rolled, what is the probability that at least one die will show a number greater than 5?
Answer:
probability is 671 out of 1296 or 51.8 %.
Step-by-step explanation:
When we roll 4 fair 6-sided dice total outcomes are
6^4 = 1296
The outcomes where no dice show greater than 5, the dice can show numbers 0,1,2,3,4,5
So the no of these outcomes where no dice show greater than 5 can be found by
5^4 = 625
No of outcomes where at least one dice will show number greater than 5 are
1296-625 = 671
Or in percentage,the probability is (671/1296)*100 = 51.8%
HELP!!!! I think its C but I'm not sure!
What does the fundamental theorem of algebra state about the equation 2x2−4x+16=0 ?
A. The fundamental theorem of algebra tells you that the equation will have two complex roots since the degree of the polynomial is 2. The roots are x=1±i7√2 .
B. The fundamental theorem of algebra tells you that the equation will have two complex roots since the degree of the polynomial is 2. The roots are x=1±i7√ .
C. The fundamental theorem of algebra tells you that the equation will have two complex roots since the leading coefficient of the equation is 2. The roots are x=1±i7√2 .
D. The fundamental theorem of algebra tells you that the equation will have two complex roots since the leading coefficient of the equation is 2. The roots are x=1±i7√ .
Answer:
The fundamental theorem of algebra tells you that the equation will have two complex roots since the degree of the polynomial is 2. The roots are [tex]x=1\pm i\sqrt{7}[/tex].
Step-by-step explanation:
Consider the provided information.
Algebra's fundamental theorem states that: Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
Now consider the provided equation.
[tex]2x^2-4x+16=0[/tex]
The degree of the polynomial equation is 2, therefore according to Algebra's fundamental theorem the equation have two complex roots.
Now find the root of the equation.
For the quadratic equation of the form [tex]ax^2+bx+c=0[/tex] the solutions are: [tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Substitute [tex]a=2,\:b=-4,\:\ and \ c=16[/tex] in above formula.
[tex]x_{1,\:2}=\frac{-\left(-4\right)\pm \sqrt{\left(-4\right)^2-4\cdot \:2\cdot \:16}}{2\cdot \:2}[/tex]
[tex]x_{1,\:2}=\frac{4\pm \sqrt{16-128}}{4}[/tex]
[tex]x_{1,\:2}=\frac{4\pm \sqrt{-112}}{4}[/tex]
[tex]x_{1,\:2}=\frac{4\pm 4i\sqrt{7}}{4}[/tex]
[tex]x_{1,\:2}=1\pm i\sqrt{7}[/tex]
Hence, the fundamental theorem of algebra tells you that the equation will have two complex roots since the degree of the polynomial is 2. The roots are [tex]x=1\pm i\sqrt{7}[/tex].
A lion hides in one of three rooms. On the door to room number 1 a note reads: „The lion is not here". On the
door to room number 2 a note reads: „The lion is here". On the door to room number 3 a note reads: „2 + 3 = 5".
Exactly one of the three notes is true. In which room is the lion?
(A) Room 1 (B) Room 2 (C) Room 3
(D) It can be in any room. (E) It is either in room 1 or room 2.
Answer: I will pick B
Step-by-step explanation:
Because it's a logical explanation
Answer:
Room 1.
Step-by-step explanation:
The note on room 3 is true. So The notes on room 1 and room 2 are untrue.
If the lion is in room 1 that would make the note on room 1 untrue - so both room 1 and room 2 notes would be untrue.
Also if the lion is in room 2 that would make both room 1 and room 2 notes true.
So the lion must be in room 1.
How many 4-inch by 4-inch squares can be cut from a rectangular piece of leather measuring 4 feet by two-thirds of a yard?
72 squares can be cut from a rectangular piece of leather measuring 4 feet by two-thirds of a yard
Solution:
Given, 4-inch by 4-inch squares can be cut from a rectangular piece of leather measuring 4 feet by two-thirds of a yard
Therefore,
Area of square = 4 x 4 = 16
Thus area of square to be cut is 16 square inches
Rectangular piece of leather measuring 4 feet by two-thirds of a yard
Convert feet to inches
1 feet = 12 inch
4 feet = 4 x 12 = 48 inches
Also,
1 yard = 36 inch
Given is a two-thirds of a yard
[tex]\frac{2}{3} \times 36 = 24\ inches[/tex]
Thus area of rectangular piece of leather is:
[tex]Area = 48 \times 24 = 1152\ square\ inches[/tex]
Total number of squares cut is given as:
[tex]\text{Total number of squares cut} = \frac{\text{area of leather}}{\text{ area of square}}[/tex]
Thus we get,
[tex]\text{Total number of squares cut} = \frac{1152}{16} = 72[/tex]
Thus 72 squares can be cut from a rectangular piece of leather measuring 4 feet by two-thirds of a yard
Answer:
72
Step-by-step explanation:
Find S25 for 1/2 + 1 + 3/2 + 2 + ...
Step-by-step explanation:
The given sequence:
[tex]\dfrac{1}{2}+1+\dfrac{3}{2}+2+ ...[/tex]
Here, first term (a) = [tex]\dfrac{1}{2}[/tex], common difference(d) =[tex]1-\dfrac{1}{2}=\dfrac{1}{2}[/tex] and
the number of terms (n) = 25
The given sequence are in AP.
To find, the value of [tex]S_{25}[/tex] = ?
We know that,
The sum of nth terms of an AP
[tex]S_{n}=\dfrac{n}{2}[2a+(n-1)d][/tex]
The sum of 25th terms of an AP
[tex]S_{25}=\dfrac{25}{2}[2(\dfrac{1}{2})+(25-1)(\dfrac{1}{2})][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[1+(24)(\dfrac{1}{2})][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[1+12][/tex]
⇒ [tex]S_{25}=\dfrac{25}{2}[13][/tex]
⇒ [tex]S_{25}=\dfrac{325}{2}[/tex]
∴ [tex]S_{25}=\dfrac{325}{2}[/tex]
Researchers wanted to know if there is a link between proximity to high-tension wires and the rate of leukemia in children. To conduct the study, researchers compared the rate of leukemia for children who lived within 1/2 mile of high-tension wires to the rate of leukemia for children who did not live within 1/2 mile of high-tension wires. The researchers found that the rate of leukemia for children near high-tension wires was higher than the rate for those not near high-tension wires. Can the researchers conclude that proximity with high-tension wires causes leukemia in children?
Answer:
COPD
Step-by-step explanation:
COPD is a lung disease caused by tobacco use and other factors
On a cross country trip the Anderson family plan to average 500 miles in 10 hours of driving each day on average how many miles per hour do the Andersons plan to drive
Answer:
50 miles per hour
Step-by-step explanation:
500 miles in 10 hours
= 500/10
= 50 miles per hour
1.95=z-2.05
Help it is so hard
Answer:
Z=4
Step-by-step
You just add 2.05 to 1.95 to get z alone
Answer: Z=4
Step-by-step explanation:
Add 2.05 and 1.95 and you get 4.
If you subtract 2.05 from 4 you get 1.95
A pizza is cut into five pieces. Four of the pieces are the same size, and the fifth size is .5 the size of each of the others. What fraction of the pizza is the smallest piece
Answer:
1/10 of pizza
Step-by-step explanation:
Let x represent size of equal pieces.
We have been given that a pizza is cut into five pieces. Four of the pieces are the same size, and the fifth size is 0.5 the size of each of the others.
This means that size of small piece would be half the size of other pieces, that is [tex]\frac{x}{2}[/tex].
Since the pizza is divided in 5 pieces, so we will divide [tex]\frac{x}{2}[/tex] by 5 as:
[tex]\frac{\frac{x}{2}}{5}=\frac{x}{2\cdot5}=\frac{x}{10}=\frac{1}{10}x[/tex]
Therefore, the smallest piece is 1/10 of the pizza.
At the county fair,animals are judged for the quality of their breeding and health.The animal pens are arranged in an array,with one animal in each pen .A barn can hold at most 10 rows of pens and at most 6 pens in each row ,with room for people to walk around them.What different ways can the planners of county fair arrange the pens for the horses and cows in the same barn? How the quantities given in the problem relate to each other?
Answer:
1 . 60!=8.31*[tex]10^{81}[/tex] ways
The rows and number of barns are related in that if we want to get the number of ways the cows and horse can be arranged
Step-by-step explanation:
At the county fair,animals are judged for the quality of their breeding and health.The animal pens are arranged in an array,with one animal in each pen .A barn can hold at most 10 rows of pens and at most 6 pens in each row ,with room for people to walk around them.What different ways can the planners of county fair arrange the pens for the horses and cows in the same barn? How the quantities given in the problem relate to each other?
if there are 10 ros and 6 barns. the number of ways animsls can be arrganged becomes
10 *6=60
60
look for 60 factorials, the number of ways
60!=8.31*[tex]10^{81}[/tex] ways
2.Permutation means arrangement . The rows and number of barns are related in that if we want to get the number of ways the cows and horse can be arranged , it makes it possible
For example find the number of ways 12 cows and 18 horses can be arranged in the barns.
we have the number of animals to be=12+18=30
60P30=60[tex]\frac{60!}{960-30)!} =\frac{60!}{30!}[/tex]
31.37*10^48 ways
Final answer:
The planners at the county fair can arrange the animal pens in various ways as long as the total number does not exceed 60, based on a maximum of 10 rows and 6 pens per row. This problem is one of combinatorics, requiring the counting and arranging of objects within given constraints.
Explanation:
The question involves arranging animal pens within a barn that can accommodate at most 10 rows of pens and up to 6 pens per row for animals like horses and cows at a county fair. This is fundamentally a problem of combinatorics, a branch of mathematics dealing with the counting, arrangement, and combination of objects. To determine the different arrangements possible for placing the pens, we would consider the combinations that do not exceed the given maximums for rows and pens per row.
The total number of pens that can fit in the barn is found by multiplying the maximum number of rows by the maximum number of pens per row, which gives us 10 rows × 6 pens per row = 60 pens as the upper limit. Therefore, the planners could arrange the pens in a variety of ways, including all rows filled with 6 pens, fewer rows with 6 pens, or more rows with less than 6 pens each, as long as the total number of pens does not exceed 60.
The quantities given in the problem relate to one another as constraints in a two-dimensional array. The planners have the flexibility to decide how many rows and pens per row to use without exceeding the maximum capacity of the structure. This scenario underscores the practical application of mathematical principles in planning and logistics.
Sam can brew 5 gallons of root beer in an hour or he can make 4 pizzas in an hour. Ben can brew 7 gallons of root beer in an hour or he can make 5 pizzas in an hour.Who has an absolute advantage in making pizza?
Answer:
Ben
Step-by-step explanation:
Ben has an absolute advantage in making of pizza because he can make five pizzas in one hour which is a greater quantity compared to Sam that makes four pizzas in one hour.
Rashaads sister gives him 2 pack of cards per month and 3 extra packs for his birthday there are 11 cards in a pack. How many cards does he get in a year?
Answer: he got 297 cards in a year.
Step-by-step explanation:
There are 12 months in a year. Rashaads sister gives him 2 pack of cards per month. This means that in a year, his sister would give him
2 × 12 = 24 packs of cards.
If there are 11 cards in a pack, then the number of cards that his sister gives him in a year would be
24 × 11 = 264 cards.
He also gets 3 extra packs for his birthday. It means that the number of cards that he gets for his birthday would be
3 × 11 = 33 cards.
Therefore, the total number of cards that he gets in a year is
264 + 33 = 297 cards
In the diagram, BC⎯⎯⎯⎯⎯∥DE⎯⎯⎯⎯⎯ .
What is AE ?
Enter your answer in the box.
___ in.
THE ANSWER IS 18in
The length of AE is [tex]18in[/tex]
Explanation:
From the figure, we can see that the two triangles ΔAED and ΔACB are similar. Thus, the legs of the triangle are proportional to each other.
Thus, we have,
[tex]\frac{EC}{DB} =\frac{AE}{AD}[/tex]
Substituting the values of the sides from the image, we get,
[tex]\frac{3}{1} =\frac{AE}{6}[/tex]
Multiplying both sides of the equation by 6, we get,
[tex]6(3)=AE[/tex]
Multiplying, we have,
[tex]18=AE[/tex]
Thus, the length of AE is [tex]18in[/tex]
Answer:
the answer is 18 inches.
Step-by-step explanation:
If a line of one billion people standing shoulder to shoulder stretches 420,334 miles what is the average shoulder width in feet of the people in line
Answer:
2.21936352 feet
Step-by-step explanation:
420334*5280 (thats feet in a mile) divided by 1000000000
Yo sup??
Average shoulder width=total lenght / number of people
since we want it in inches therefore
Final answer=Average shoulder width*63360
=420334*63360/1,000,000,000
=26.62 inches
Hope this helps
Below is the five-number summary for 136 hikers who recently completed the John Muir Trail (JMT). The variable is the amount of time to complete the 212-mile hike from Yosemite Valley across the high Sierras to the top of Mount Whitney. Five-number summary: Minimum: 9 days Q1: 18 days Median: 21 days Q3: 28 days Maximum: 56 days If we use the 1.5 * IQR rule to determine whether there are any outliers, what is the right boundary?
Answer:
43
Step-by-step explanation:
We have the following data:
Total number of hikers: 136
Minimum: 9 days
Q1 : 18 days
Median: 21 days
Q3: 28 days
Maximum: 56 days
Using the 1.5 Interquartile rule means:
Left boundary: Q1 - 1.5 × IQR
Right boundary: Q3 + 1.5 × IQR
We first calculate the IQR (Interquartile Range): Q3 - Q1
⇒ 28 - 18 = 10
Right boundary: 28 + 1.5 × 10
= 28 + 15
= 43
Hence the right boundary is 43.
A foot path of uniform width runs all around inside of a rectangle field 45m long and 36m wide.If the area of the path is 234 m , find the width of the path
Answer:
The width of path is 1.5 m
Step-by-step explanation:
We are given the following in the question:
Field:
Length = 45 m
Width = 35 m
Area of filed =
[tex]\text{Area} = \text{Length}\times \text{Width} = 45\times 36 = 1620[/tex]
The area of rectangular field the is 1620 square meter.
Area of path = 234 m
Let x be the width of path.
Area of field without path = 1620 - 234 = 1386 square meter.
Now, dimensions of field without path is:
Length = [tex]45 -x - x = 45 -2x[/tex]
Width = [tex]36 -x - x = 36 - 2x[/tex]
[tex]\text{Area} = \text{Length}\times \text{Width}[/tex]
Thus, we can write:
[tex](45-2x)(36-2x) = 1386[/tex]
[tex]1620 - 90x - 72x + 4x^2 = 1386\\4x^2 - 162x + 234 = 0\\2x^2 - 81x + 117 = 0\\(2x - 3)(x - 39) = 0\\x = 1.5, x = 39[/tex]
We cannot take the width as 39 m, thus, the width of path is 1.5 m.
When using rational expectations, forecast errors will, on average, be ________ and ________ be predicted ahead of time. A) zero; cannot B) negative; can C) positive; can D) positive; cannot
Answer:
A) zero; cannot
Step-by-step explanation:
In line with the principle of rational expectations, expectation errors are unpredictable. The expectations of all available information will not differ from the optimal projections.The word optimal projection is inexorably intertwined with the best guess in rational expectations theory.
Please help me.
Rectangles F and H are similar. If rectangle F has dimensions of 5x10 and rectangle H has dimensions of 15 by an unknown amount. What is the unknown dimension?
I tried everything, even looking in that useless mathbook, I'm resorting to brainly as a last hope.
Since they are similar both dimensions would have the same ratio. The ratio of 5 and 15 is 3. 15 is 3 times larger than 5, so the unknown dimension is 3 times larger than the known dimension.
3 x 10 = 30
The unknown dimension is 30
In a certain furniture store, each week Nancy earns a salary of $240 plus 5% of the amount of her total sales that exceeds $800 for the week. If Nancy earned a total of $450 one week, what were her total sales that week ?
A. $2,200
B. $3,450
C. $4,200
D. $4,250
E. $5,000
Answer:
$4200
Step-by-step explanation:
450 - 240 = 210 this is the amount of her commissions for the week.
we are looking for x = sales for the week
x * .05 = 210
x = 210/.05
x = 4200
Answer:c 4200
Step-by-step explanation:
Find the area of the figure.
Answer:
20 square units
Step-by-step explanation:
The figure shows a triangle whose;
Base AC is 8 units Height is 5 unitsWe are supposed to get its area;
Area of a triangle is given by the formula;
Area = 0.5×b×h
Thus;
Area = 0.5 × 8 units × 5 units
= 20 square units
Hence, the area of the figure is 20 square units
The box plots show the target heart rates of men 20–40 years old and men 50–70 years old. Which statement is best supported by the information in the box plots?
Your Question is incomplete, here is the complete statement of the question with the box plots in the attached file.
Question statement:
The box plots show the target heart rates of men 20-40 years old and men 50-70 years old.
Which statement is best supported by the information in the box plots?
A)
The range of the data for men 20-40 years old is less than the range of the
data for men 50-70 years old.
B)
The median of the data for men 20-40 years old is less than the median of
the data for men 50-70 years old.
o
The minimum target heart rate for men 20-40 years old is less than the
minimum target heart rate for men 50-70 years old.
D)
The interquartile range of the data for men 20-40 years old is greater than
the interquartile range of the data for men 50-70 years old.
Answer:
D
Step-by-step explanation:
please find the box plots in the file attached below.
looking at the box plots we can say that the answers is D due to following reasons:
Option A is incorrect:
The range of the data for men 20-40 years old is not less than the range of data for men 50-70 years old because for men 20-40 years old range is 80 and for men 50-70 years old range is 70.
Option B is incorrect:
The median of the data for men 20-40 years old is not less than the range of data for men 50-70 years old because for men 20-40 years old median is 130 and for men 50-70 years old median is 110.
Option C is incorrect:
The minimum target heart rate for men 20-40 years old is not less than the minimum target heart rate for men 50-70 years old because for men 20-40 years old the minimum target heart rate is 90 and for men 50-70 years old the minimum target heart rate is 75.
Option D is correct:
The interquartile range of the data for men 20-40 years old is greater than the interquartile range of data for men 50-70 years old because for men 20-40 years old interquarile range is [tex]Q_{3}-Q_{1}=152.5-107.5=45[/tex] and for men 50-70 years old interquartile range is [tex]Q_{3} -Q_{1} =130-90=40[/tex].
Charlie has the utility function u(xa, xb) =xaxb. If Charlie's income is $40, the price of apples is $4, and the price of bananas is $2, how many apples are there in the best bundle Charlie can afford?
Answer:
There are 5 apples in the best bundle Charlie can afford
Step-by-step explanation:
If the utility function is u(xa, xb) , where xa represent the quantity of apples and xb is the quantity of bananas then we want to choose the quantity of bananas and apples that maximises the utility of Charlie for the same budget restriction ( get the most benefit for the same money).
The budget restriction is
$4* xa + $2* xb = $40
then
u(xa, xb) =xa*xb
4*xa + 2*xb = 40 → xb = (40 - 4*xa)/2 = 20 - 2*xa
replacing in the utility function
u(xa, xb) =xa* (20 - 2*xa) = 20*xa - 2*xa²
the maximum of this function is obtained when the derivative of the utility function with respect to xa is 0 . Thus
du/dxa = 20 - 4*xa = 0 → xa = 20/4 = 5
then for
xa=5 apples
xb=20 - 2*xa = 20 - 2*5 = 10 bananas
Charlie maximises his utility . Therefore there are 5 apples in the best bundle Charlie can afford
To determine the best bundle Charlie can afford, we need to maximize his utility function u(xa, xb) = xaxb subject to his income and the prices of apples and bananas. The maximum value of xa represents the number of apples in the best bundle Charlie can afford, which is 5.
Explanation:To determine the best bundle Charlie can afford, we need to maximize his utility function u(xa, xb) = xaxb subject to his income and the prices of apples and bananas. Let's denote the quantity of apples as xa and the quantity of bananas as xb. Since Charlie's income is $40, we have 4xa + 2xb ≤ 40. Using this inequality, we can find the maximum value of xa, which represents the number of apples in the best bundle Charlie can afford.
To solve for xa, we rearrange the inequality:
4xa ≤ 40 - 2xbxa ≤ (40 - 2xb) / 4Now, let's look at the utility function u(xa, xb) = xaxb. To maximize the utility function, we take the derivative of u with respect to xa and set it equal to zero.
∂u/∂xa = 0(d/da)(xa * xb) = 0xb = 0 / xaSince xb is in the denominator, its value should be greater than zero. Therefore, the best bundle Charlie can afford will have the maximum value of xa such that 4xa + 2(could be anything greater than 0) ≤ 40. Solving for xa, we get xa ≤ 5.
Hence, Charlie can afford a maximum of 5 apples in the best bundle.
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