The length of the rectangle is 1.054 units and the width of the rectangle is 4.72 units.
Length times width equals the area of a rectangle. Let's assume that x represents length and y represents width. To determine the dimensions of the rectangle, we must apply y values to the area formula, differentiate with respect to x, and then set differentiate to zero.
Parabola: [tex]y=5-x^{2}[/tex]
The equation of parabola indicates that it opens down and the axis of symmetry is x=0. So we can take the vertices of the rectangle on the x-axis as [tex]A=(-x,0)[/tex] and [tex]D=(x,0)[/tex]). The other two vertices above the x-axis on the parabola will be [tex]B=(-x, 5-x^{2} )[/tex] and [tex]C=(x,5-x^{2} )[/tex]
Dimensions of the rectangle will be,
[tex]x-(-x)=2x[/tex] and [tex](5-x^{2} )[/tex]
Area of the rectangle (A) = [tex]2x(5-x^{2} )[/tex]
Area function when differentiated (A′) = [tex]2x(-2x)+(5-x^{2} )(2)[/tex]
A′= [tex]-4x^{2} + 10 -2x^{2}[/tex]
A′= [tex]-6x^{2} +10[/tex]
Critical points for maxima and minima ⟹ [tex]-6x^{2} +10=0[/tex]
⟹ [tex]10=6x^{2}[/tex]
⟹ [tex]x^{2} = \frac{10}{6}[/tex]
⟹ [tex]x= \sqrt \frac{10}{6}[/tex]
⟹ [tex]x=[/tex] ±[tex]0.527[/tex]
Maximum area of 2X2 rectangle(square)
A= [tex]2*0.527(5- 0.527^{2} )[/tex] = [tex]4.97[/tex] [tex]units^{2}[/tex]
Length = 2x = 1.054 units
Width = [tex](5-x^{2} )[/tex] = 4.72 units
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. DIG DEEPER What is Descartes's number?
• It is less than 300.
• It is greater than 200.
• The ones digit is 2 more than the hundreds digit.
• The tens digit is 2 more than the ones digit.
●
Descartes's number is the 46.989, which is less than 47 and higher than 46.98.
Define Descartes's number.A Descartes number is an odd number with the formula n = m p, where m and p are coprime numbers and 2n = (m) (p + 1). P is regarded as a "spoof" prime in this formula. The only known example is the one that was provided. The invention of Cartesian or analytic geometry, which employs algebra to describe geometry, is one of Descartes's most enduring contributions. In equations, Descartes "introduced the convention of denoting unknowns by x, y, and z, and knowns by a, b, and c." A Descartes number is an odd integer that, if one of its composite elements were prime, would have been an odd perfect number.
Given,
Descartes's number is,
46.989, which is less than 47 and higher than 46.98.
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Farah wants to have a cool million dollars to retire with when she is 55. How much must she invest today at age 19 into an account that pays 8% compounded monthly to accomplish this goal?
Sam also wants a cool million dollars when he is 55 but is willing to put away money every year. How much must he put away each year after his 21st birthday with same interest rate as Farah but his investment is compounded annually?
To calculate how much Farah must invest today at age 19 to have a cool million dollars when she is 55, we need to know how many years she has until she retires. Since Farah wants to retire at age 55, she has 55-19 = <<55-19=36>>36 years until she retires.
We also need to know the interest rate that the account pays and the frequency at which the interest is compounded. In this case, the interest rate is 8% and the interest is compounded monthly.
To calculate how much Farah must invest, we can use the formula for the future value of an annuity, which is given by:
FV = PMT * (((1 + i)^n - 1) / i)
where FV is the future value that Farah wants to have, PMT is the amount that Farah must invest each period, i is the interest rate per period, and n is the total number of periods.
Since Farah wants to have a cool million dollars, we can set FV equal to $1,000,000. We also know that the interest rate is 8% and that the interest is compounded monthly, so we can set i equal to 8% / 12 = 0.006667. Finally, we know that Farah has 36 years until she retires, so we can set n equal to 36 * 12 = 432, since there are 12 months in a year.
Substituting these values into the formula above, we get:
FV = PMT * (((1 + i)^n - 1) / i)
FV = PMT * (((1 + 0.006667)^432 - 1) / 0.006667)
To solve for PMT, we can divide both sides of the equation by ((1 + 0.006667)^432 - 1) / 0.006667, which gives us:
PMT = FV / (((1 + i)^n - 1) / i)
PMT = $1,000,000 / (((1 + 0.006667)^432 - 1) / 0.006667)
Calculating this value gives us a result of PMT = $176.72, which is the amount that Farah must invest each month to have a cool million dollars when she is 55.
Now, let's consider Sam's situation. Sam also wants to have a cool million dollars when he is 55, but he is willing to put money into his account every year instead of every month. This means that the frequency at which the interest is compounded will be annual instead of monthly.
To calculate how much money Sam must put into his account each year, we can use the same formula as above but with a few changes. In particular, we need to set i equal to 8% / 1 = 0.08, since the interest is compounded annually, and we need to set n equal to 55-21 = <<55-21=34>>34, since Sam has 34 years until he retires.
Substituting these values into the formula above, we get:
FV = PMT * (((1 + i)^n - 1) / i)
FV = PMT * (((1 + 0.08)^34 - 1) / 0.08)
Solving for PMT gives us:
PMT = FV / (((1 + i)^n - 1) / i)
PMT = $1,000,000 / (((
(2/15x - 5) - (-0.7x + 2)
The expression (2/15x - 5) - (-0.7x + 2) is simplified to 5/12x - 3
What is an algebraic expression?An algebraic expression can be defined as an expression made up of variables, terms, coefficients, factors and constants.
These expressions are also made up of arithmetic operations.
From the information given, we have;
2/15x - 5) - (-0.7x + 2)
expand the bracket
2/15x - 5 + 7/10x - 2
collect like terms
2/15x + 7/10x - 5 + 2
Add the like terms
4x + 21x /30 - 3
Add the numerators
25x/60 - 3
Divide the common terms, we have;
5/12x - 3
Hence, the expression is 5/12x - 3
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Journalizing, adjusting, adjusted trial balance
Harrison’s Repair Service shows opening balance for some accounts on 01 January 20X1: Cash: $36,000; Accounts Receivable: 60,500; Prepaid Rent: $2,400; Supplies: $2,500; Equipment: $13,900; Accumulated depreciation - Equipment: $3,150; Accounts Payable: $2,780; Notes Payable: 16,000; Unearned service revenue: 6,120; Share capital: 86,915; Retained Earnings: 15,335; Dividends: 15,000.
The following transactions in January were:
January 1 Paid the monthly rental fee, $790.
1 Made the monthly payment to Apple Company, $1,700 in order to decrease Accounts
Payable.
6 Purchased additional repair supplies on credit from Pineapple Company, $1820.
15 Received cash for repair service performed, $2,950.
20 Paid cash for an advertisement in the local newspaper, $370.
23 Paid Pineapple Company on account, $1,200
30 Collected payment for repair fees earned, $7,460.
30 Recorded a dividend paid to stockholders, $8,100.
Required
A, Prepare journal entries to record the January transactions.
B, Using the following information, record adjusting entries in the general journal and prepare the T- accounts for all accounts
1. One month of rent has expired. The rental policy is valid in one year starting from the first day of November 20X0.
2. The inventory of unused repair supplies is $1,520
3. The estimated depreciation on repair equipment is $350.
4. Accrued one-month interest expense on Note Payable that will be paid on February 1 (Borrowing Note was in 6-month; annual interest rate is 6%).
5. Accrued salaries for 2 employees at the end of the month, payment for each person is $500
6. Service Revenue still unearned at the end of the period is $3,675
7. Service revenue earned but not billed is $1,800
C. Prepare an adjusted trial balance from above T-accounts
Answer:
Step-by-step explanation: A. Prepare journal entries to record the January transactions.
Paid the monthly rental fee, $790.
Debit Credit
Rent Expense 790
Cash 790
Made the monthly payment to Apple Company, $1,700 in order to decrease Accounts Payable.
Debit Credit
Accounts Payable 1,700
Cash 1,700
Purchased additional repair supplies on credit from Pineapple Company, $1,820.
Debit Credit
Supplies 1,820
Accounts Payable 1,820
Received cash for repair service performed, $2,950.
Debit Credit
Service Revenue 2,950
Cash 2,950
Paid cash for an advertisement in the local newspaper, $370.
Debit Credit
Advertising Expense 370
Cash 370
Paid Pineapple Company on account, $1,200.
Debit Credit
Accounts Payable 1,200
Cash 1,200
Collected payment for repair fees earned, $7,460.
Debit Credit
Accounts Receivable 7,460
Cash 7,460
Recorded a dividend paid to stockholders, $8,100.
Debit Credit
Dividends 8,100
Cash 8,100
B. Using the following information, record adjusting entries in the general journal and prepare the T- accounts for all accounts
One month of rent has expired. The rental policy is valid in one year starting from the first day of November 20X0.
Debit Credit
Prepaid Rent 90
Rent Expense 90
The inventory of unused repair supplies is $1,520
Debit Credit
Supplies 1,520
Supplies Expense 1,520
The estimated depreciation on repair equipment is $350.
Debit Credit
Depreciation Expense 350
Accumulated Depreciation - Equipment 350
Accrued one-month interest expense on Note Payable that will be paid on February 1 (Borrowing Note was in 6-month; annual interest rate is 6%).
Debit Credit
Interest Expense 100
Interest Payable 100
Accrued salaries for 2 employees at the end of the month, payment for each person is $500
Debit Credit
Salaries Expense 1,000
Salaries Payable 1,000
Service Revenue still unearned at the end of the period is $3,675
Debit Credit
Unearned Service Revenue 3,675
Service Revenue 3,675
Service revenue earned but not billed is $1,800
Debit Credit
Service Revenue 1,800
Accounts Receivable 1,800
C. Prepare an adjusted trial balance from above T-accounts
Debit Credit
Cash 35,080
Accounts Receivable 63,080
Prepaid Rent 1,510
Supplies 1,300
Equipment 13,900
Accumulated Depreciation - Equipment 3,500
Accounts Payable 2,380
Notes Payable 16,100
Interest Payable 100
Salaries Payable 1,000
Unearned Service Revenue 3,675
Service Revenue 6,475
Rent Expense 870
You have at most $23 to spend at a market. Apples cost $0.50 each, and pears cost $0.75 each. Let a be numbers of apples and p be numbers of pears. Write an inequality that represents the numbers of apples and pears you can afford
Do not include the dollar sign ($) in your inequality.
Answer:
0.5a + 0.75p ≤ 23
Step-by-step explanation:
Cost of a apples: 0.5a
Cost of p pears: 0.75p
The total cost must be $23 or less.
0.5a + 0.75p ≤ 23
Given the coordinates (0, 0) and (4, 1), the distance is
Answer:
distance = √17
Step-by-step explanation:
distance = √(y2 - y1)^2 + (x2 - x1)^2
distance = √(1 - 0)^2 + (4 - 0)^2
distance = √17
I hope my answer helps you.
A survey asked people what alternative transportation modes they use. The results are below:
16% of people answered light rail.
10% of people answered bus.
11% of people answered carpool.
2% of people answered carpool and bus.
6% of people answered light rail and carpool.
3% of people answered light rail and bus.
2% of people answered all three.
Fill in the following Venn Diagram with the percents that each region represents then answer the
questions below.
light rail
IV.
11.
VII.
bus
%
VI.
carpool
III.
What percent of people answered light rail or bus, but not carpool? 15
What percent of people answered carpool and bus?
What percent of people did not answer light rail, carpool, or bus?
y
What percent of people answered exactly one of the options?
What percent of people answered light rail, carpool, or bus?
VIII.
%
%
%
U
%
Fill in the Venn Diagram and answer the questions.
Thank you.
The completed Venn Diagram is given by the image shown at the end of the answer.
The percentages are given as follows:
Light rail or bus, but not carpool: 10%.Carpool and bus: 0%.Did not answer light rail, carpool or bus: 72%.Exactly one: 21%.Light rail, carpool, or bus: 28%.How to fill the Venn Diagram?The Venn diagram starts to be filled from the most restrictive conditions, towards the least restrictive.
2% of people answered all three, hence:
V = 2%.
3% of people answered light rail and bus, hence:
IV + V = 3%
IV = 1%.
6% of people answered light rail and carpool, hence:
II + V = 6%
II = 4%.
2% of people answered carpool and bus, hence:
V + VI = 2%
V = 0%.
11% of people answered carpool, hence:
III + II + V + VI = 11
III + 4 + 2 + 0 = 11
III = 5%.
10% of people answered bus, hence:
IV + V + VI + VII = 10
1 + 2 + 0 + VII = 10
VI = 7%.
16% of people answered light rail, hence:
I + II + IV + V = 16
I + 4 + 2 + 1 = 16
I = 9%.
Then the or percentage is given as follows:
9 + 7 + 5 + 1 + 2 + 4 = 28%.
The percentage that uses other mediums is given as follows:
VIII = 100 - 28 = 72%.
Then the missing percentages are given as follows:
Light rail or bus, but not carpool: I + IV = 9 + 1 = 10%.Carpool and bus: VI = 0%.Exactly one: I + III + VII = 9 + 5 + 7 = 21%.More can be learned about Venn Diagrams at https://brainly.com/question/24713052
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A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time on death row could likely be 15 years. Is this a test of one mean or proportion? State the null and alternative hypotheses. H_o: H_a: Is this a right-tailed, left-tailed, or two-tailed test? What symbol represents the random variable for this test? In words, define the random variable for this test. Is the population standard deviation known and, if so. what is it? Calculate the following: Which test should be used? State the distribution to use for the hypothesis test. Find the p -value. Al a pre -conceived a = 0.05, what is your Decision: Reason for the decision: Conclusion (write out in a complete sentence):
At the null hypothesis, it is tested if the mean is of 15 years, that is:
[tex]H_0: \mu = 15[/tex]
At the alternative hypothesis, it is tested if the mean is different of 15 years, that is:
[tex]H_1: \mu \neq 15[/tex]
What is the test statistic?The equation that gives the test statistic is defined as follows:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.The values of these parameters in this problem are given as follows:
[tex]\overline{x} = 17.4, \mu = 15, s = 6.3, n = 75[/tex]
Hence the test statistic is of:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{17.4 - 15}{\frac{6.3}{\sqrt{75}}}[/tex]
t = 3.30.
What is the p-value?Considering a two-tailed test, as we are testing if the mean is different of a value, with t = 3.30 and 75 - 1 = 74 df, the p-value of the test is of:
0.0015.
Since the p-value is less than the significance level of 0.05, the null hypothesis is rejected, meaning that there is enough evidence to conclude that prisoners spend a time different of 15 years in the death row.
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A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the height, to the nearest foot, at a time of 3 seconds.
The height of the rocket in given time using regression equation is 586 feet.
The regression equation is, y = 7.70961x² + 176.30826x - 12.44515
Given,
A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet
That is,
x ; 0.6, 1.3, 1.8, 2.5, 3.1
y ; 108, 218, 285, 368, 436
We have to write a quadratic regression equation for the given set;
The quadratic regression equation is;
y = 7.70961x² + 176.30826x - 12.44515
Now,
We have to find the height at a time of 3 seconds using the equation;
So,
y = 7.70961 x 3² + 176.30826 x 3 - 12.44515
y = 69.38649 + 528.92478 - 12.44515
y = 585.86612
y = 586
Height of the rocket= 586 feet
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How factorize this : ( + )( – ) + (– + )( – )
Answer:
this very easy to factorize this because (-+)is- (+-)-
(++)+ (--)+ but the sign which is bigger we have to keep that
a complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations among them. this information is needed to solve oscillation problems of this type. the position of a 60 gg oscillating mass is given by x(t)
a complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations then the position of a 60 gg oscillating mass is given by x(t) -20 sin 10t
this information is needed to solve oscillation problems of this type
the position of a 60 gg oscillating mass is given by x(t)
x = 2 cos wt = 2 cos 10t ; w = 10
velocity = dx/dt = -2 x 10 sin 10 t.=- 20 sin 10t
t = .4
velocity = -20 sin 10 x .4 = -20 sin 4 = -20 x -0.7568 = 15.136 cm /s
w = √ k / m = 10 = √ k / .05
k = 15.136 N/m
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Solve
7) 6-4(6 n+7) ≥122
The solution of the given inequality 6 - 4( 6 n+7 ) ≥ 122 is given by
n ≤ -6.
As given in the question,
Given inequality is :
6 - 4( 6 n+7 ) ≥ 122
Open the parenthesis of the given inequality we get,
⇒ 6 - 24n - 28 ≥ 122
⇒ 6 - 28 -24n ≥ 122
⇒ -22 -24n ≥ 122
Add 22 on both the side of inequality we get,
⇒ -22 + 22 -24n ≥ 122 + 22
⇒ -24n ≥ 144
Divide both the side by ( -24 ) we get,
⇒ n ≤ -144/ 24
⇒ n ≤ -6.
This shows that n has all the less than or equal to -6.
Therefore, the solution of the given inequality is equal to n ≤ -6.
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the president of blitz sales enterprises sells kitchen products through cable television infomercials. he gathered data from the last 15 weeks of sales to determine the relationship between sales and the number of infomercials. infomercials sales ($000s) 20 3.2 15 2.6 25 3.4 10 1.8 18 2.2 18 2.4 15 2.4 12 1.5 22 2.5 15 2.4 25 3.0 16 2.7 12 2.0 20 2.6 25 2.8
The regression equation is Y=1117.84+73.62X for the data as follows:
(20,3200), (15,3600), (25,3400), (10,1800), (18,2200), (18,2400), (15,2400), (12,1500), (22,2500), (15,2400), (25,3000), (16,2700), (12,2000), (20,2600), (25,2800)
where (x,y) denotes infomercials and sales, respectively.
According to regression equation Y=a+bX,
a = ((Σy) x (Σx^2) - (Σx) x (Σxy)) / (n x (Σx^2) - (Σx)^2)
a = ((36500 x 5126)-(268 x 677000)) / ((15 x 5126)-(268 x 268))
a = 1117.84
and, b = (n(Σxy)-(Σx)(Σy)) / (n(Σx^2)-(Σx)^2)
b = ((15 x 677000)-(268 x 365000)) / ((15 x 5126)-(268 x 268))
b = 73.62
Putting the values of a and b in the regression equation, we get the desired equation Y=1117.84+73.62X.
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Find the composite function
Please help me! I'm so confused and awful at math
Step-by-step explanation:
it is really a simple concept : instead of the isolated variable x you grab the whole function of g(x) and put that into the place of any x appearance in f(x) : g(x) becomes the input argument for f(x).
so,
f(g(x)) = sqrt(36/(x + 3)) = 6×sqrt(1/(x + 3))
what is the arithmetic mean of the altitude lengths of an isosceles triangle with two sides of length 13 cm and one side of length 10 cm? Express your answer as a common fraction.
The arithmetic mean of the three altitudes is equal to 10.1533 or 1523/150
What is the arithmetic mean of a set of observations?The Arithmetic Mean (AM) or called average is the ratio of the sum of all observations to the total number of observations and is given by [tex]A= \frac {1}{n} \sum \limits_{i=1}^n a_i[/tex]
A = arithmetic mean
n = number of values
a₁ = data set values
Given here length of two equal sides is 13 cm and the other side is 10cm
let the Δ be ΔABC with AC=AB= 13 & BC=10cm and the altitudes on sides AC, AB, and BC with the midpoints P, Q, and R be AP, BQ, and CR respectively than by Pythagoras theorem we get AP = √13²-5²
= 12
As the altitude of an isosceles triangle bisects the base in this case BC
Then by area of ΔABC is = 1/2×12×10
= 60 cm²
Again BQ is an altitude therefore area ⇒ 60=1/2×BQ×13
⇒BQ=9.2307cm
Similarly, we can find CR = 9.2307 cm
Thus Arithmetic mean of the altitudes is =( 9.2307+9.2307+12)/3
=10.1533 cm
= 1523/150 cm
Hence, the answer as a common fraction is 1523/150
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A group of 2342 students were surveyed about the courses they were taking at their college with the
following results:
1031 students said they were taking Dance.
1058 students said they were taking Science.
1059 students said they were taking English.
449 students said they were taking Science and Dance.
459 students said they were taking Science and English.
456 students said they were taking Dance and English.
218 students said they were taking all three courses.
Fill in the following Venn Diagram with the cardinality of each region then answer the questions
below.
Science
IV.
11.
VII.
V.
English
VI.
How many students took Science, Dance, or English?
How many students took Dance and English, but not Science?
How many students took Science or didn't take Dance?
How many students took none of the courses?
Dance
III.
How many students took Science or English, but not Dance?
VIII.
Hello,
I hope you and your family are doing well!
To find the number of students who took Science, Dance, or English, you can add the number of students in each individual category and then subtract the number of students who took all three courses, since those students were counted twice. This gives a total of 1031 + 1058 + 1059 - 218 = 3100 students.
To find the number of students who took Dance and English, but not Science, you can add the number of students in region VII and region VIII, which is 456 + 11 = 467 students.
To find the number of students who took Science or didn't take Dance, you can add the number of students in region I and region III, which is 1031 + 218 = 1249 students.
To find the number of students who took none of the courses, you can add the number of students in region II and region IV, which is 449 + 459 = 908 students.
To find the number of students who took Science or English, but not Dance, you can add the number of students in region V and region VI, which is 1059 + 459 = 1518 students.
----
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Happy Holidays!
find a cubic function, in the form below, that has a local maximum value of 4 at -4 and a local minimum value of 0 at 2.f (x)
The cubic function is f(x) = x^3 + 8x^2 - 32x + 32, which has a local maximum value of 4 at x = -4 and a local minimum value of 0 at x = 2.
The cubic function f(x) is of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. In this case, a = 1, b = 8, c = -32, and d = 32. The local maximum value of 4 occurs when x = -4 and the local minimum value of 0 occurs when x = 2.To find the equation, we must use the given information to solve for the constants. The function must equal 4 when x = -4, so f(-4) = 4. Plugging in the values for the constants we have 4 = -4^3 + 8(-4)^2 - 32(-4) + 32. Solving this equation, we find that d = 32, so the equation becomes f(-4) = -64 + 256 + 128 + 32 = 4. Similarly, we set f(2) = 0, so 0 = 8 + 16 - 64 + 32. Solving this equation, we find that c = -32, so the equation becomes f(2) = 8 + 16 - 32 + 32 = 0. Finally, we have the equation f(x) = x
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find a cubic function, in the form below, that has a local maximum value of 4 at -4 and a local minimum value of 0 at 2.f (x) = x^3 + 8x^2 - 32x + 32
Compute the scalar surface integral ∬T(z+y)dS where T is the triangle (including its interior) with vertices (0,0,2), (0,1,0)( and (1,0,0)
the scalar surface integral ∬T(z+y)dS where T is the triangle (including its interior) with vertices (0,0,2), (0,1,0)( and (1,0,0) is 1
We can compute the scalar surface integral ∬T(z+y)dS using the parametric form of the surface integral.
Let S be the surface of the triangle. We can parameterize S as follows:
S(u,v) = (u, v, 2-u-v)
where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1-u.
Then, the scalar surface integral ∬T(z+y)dS is defined as:
∬T(z+y)dS = ∬S(u,v) (2-u-v + v) dudv
= ∫0→1∫0→1-u (2-u-v + v) dudv
= ∫0→1 [2u + (1-u)2 - (1-u)2u - u2(1-u)] du
= ∫0→1 (2u + 1 - 2u2 - u3) du
= [u2 + u - u4/4]0→1
= 1/4 + 1 - ¼
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Dec 09, 1:25:07 PM Given the following exponential function, identify whe growth or decay, and determine the percentage rate of Growth y = 2400(1.78)* % increase Submit Answer
The exponential function is y = 2400(1.78)x
Since 1.78 < 1, the function is an exponential decay function.
The function can also be written as 2400(1 - 0.04)x.
The rate of decrease is 4% (that's 0.04 expressed as a percent)
What is exponential function?The exponential function is a mathematical function denoted by f(x)=\exp or e^{x}. Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".To learn more about algebras refer to:
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What is the position of E on the number line below?
Write your answer as a fraction or a mixed number.
E
The position of E as a mixed number is [tex]2\frac{4}{5}[/tex].
What is number line?
A number line is a diagram of a graduated straight line used to represent real numbers in elementary mathematics. It is assumed that every point on a number line corresponds to a real number, and that every real number corresponds to a point.
Let on a number line the position of E is given.
We have to find E as a mixed number.
As we can see there are 5 parts between 2 and 3.
So each part is 0.2 cm.
The position of E is in fourth part.
So, 4 x 0.2 = 0.8 cm
2 + 0.8 = 2.8 is the position of E.
2.8 can be written as.
28/10 = 2 8/10 = 2 4/5
Hence, the position of E as a mixed number is [tex]2\frac{4}{5}[/tex].
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Ben bought 20 shares of stock at $10.00 per share and sold them for $12.00 per share.
What was his ROI (Return on Investment)?
Alice has shared that her RSA public key is n = 33, e = 7. Her private key is d = 3. She was sent the encrypted number 21. Decrypt the number.
Answer: The encrypted number is 25 and this can be determined by using the formula from RSA algorithm.
Given :
Alice has shared that her RSA public key is n = = 33, e = 7.
The formula from the RSA algorithm is used to encrypt the number 16. The formula is given by:
where e = 7, n = 33 and m = 16.
Now, substitute the known values in the above equation.
Now, by using the modulo function the above expression becomes:
c = 25
So, the encrypted number is 25.
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Step-by-step explanation:
)) A football team has $630 to spend on hats for their fans. Each hat costs $5. How many
hats can the team buy?
You can spend at most $3.50 at a market. Apples cost $0.40 each, and pears cost $0.65 each. Let x be numbers of apples and y be numbers of pears. Write an inequality that represents the numbers of apples and pears you can buy. Is (3,4) a solution of the inequality?
Answer:
Step-by-step explanation:
.40x + .65 ≤3.50
(3, 4) I think it is a solution to the inequality. Because when you graph it (3, 4) looks like it is shaded meaning that it is a solution. (If wrong, I'm sorry)
The inequality that represents the numbers of apples and pears you can buy is 0.40x + 0.65y ≤ 3.50.
What is a solution set to an inequality or an equation?If the equation or inequality contains variable terms, then there might be some values of those variables for which that equation or inequality might be true. Such values are called solution to that equation or inequality. Set of such values is called solution set to the considered equation or inequality.
Given;
The cost of the apples = $0.40 per apple
So, the cost of x apples is 0.40x.
The cost of the pears = $0.65 per pear
So, the cost of y pears is 0.65y.
The total cost of the apples and pears is the sum of the cost of the apples and the cost of the pears, which is 0.40x + 0.65y;
Since we can spend at most $3.50 at the market, the total cost of the apples and pears must be less than or equal to $3.50;
Therefore, the inequality that represents the numbers of apples and pears you can buy will be 0.40x + 0.65y ≤ 3.50;
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Let f (h) be the number of riders on a train h hours after 9 A.M. Explain the meaning of the
statement f (3) = 50.
Answer:
At 12:00 there were 50 riders
Step-by-step explanation:
3 hours after 9 would be noon. At that time there are 50 riders
what is function for 3x-5
In each of 10 and 11, show that the given system has no periodic solutions other than constant solutions. dx dt = 2x 3y xy2, dy dt = y + x3 x2 y
The system does not have a periodic solution in each of 10 and 11.
A solution of the period depends on the independent variable t is called Periodic Solution. For a periodic solution x(t)( in the case of a system, x is a vector), there is a number T≠0 such that
x(t+T)=x(t) for t∈R.
The given equations are:
[tex]\frac{dx}{dt} =-2x-3y-xy^2\\\\\frac{dx}{dt} =y+x^3+x^2y[/tex]
F is the derivative of dx/dt and Gis the derivative of dy/dt:
[tex]F(x,y)=-2x-3y-xy^2\\G(x,y)=y+x^3+x^2y[/tex]
When Fx+Gx has the same sign on the entire domain D, then there are no periodic solutions on the domain D.
Fx+Gx=-2-y^2+1-x^2
=-1-y^2-x^2
<for all real numbers x,y[tex]\neq[/tex]0
Fx+Gx have the same sign for all pairs of real numbers with x[tex]\neq[/tex]0 and y[tex]\neq[/tex]0
Thus, there are no periodic solutions in the system(excluding contact solutions)
Hence proved.
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Problem 5: (8 marks) Find the Taylor series for the following function, centred at the given a.
(a) f(x) = 7 cos (−x), a = 0.
(b) f(x) = x^4 + x^2 + 1, a = −2.
(c) f(x) = 2^x, a = 1.
(d) f(x) = x tan^−1(x^2), a= 0.
Taylor Series for following parts,
a) f(x) = 7 - (7/2)x² + (7/24)x⁶ +......
b)f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......
c)f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3 (log2)³(x-1)³ + --------
d)f(x) = x² + ----
What is Taylor Series ?The Taylor series giving the expansion of the function f(x) in the neighborhood of the point a. If the function is continuous in the neighborhood, all its derivatives exist, and the series converges, then The form f(x) = f(a)+f′(a)/1!(x−a)+(f′′(a)/2!)(x−a)²+⋯+(fⁿ(a)/n! )(x−a)ⁿ --(1) where fⁿ(a) is the nᵗʰ derivative of f(x) evaluated at a.
a) f ( x) = 7 cos(-x) = 7 cos(x) , a = 0
f(a ) = 7 cos(0°) = 7
f'(x) = - 7 sin(x) , f'(a) = 7 sin(0°) = 0
f"(x) = - 7cos(x) , f"(a) = -7 cos(0°) = -7
f"'(x) = 7 sin(x) , f"'(a) = 7 sin(0° ) = 0
....................................
put all the values in above equation (1) we get,
f(x) = 7 + 0/1! (x -0) - (7/2!)(x-0)² )+ 0 +(7/4!) (x-0)⁴-0 + ----------
f(x) = 7 - (7/2)x² + (7/24)x⁶ +......
b) f(x) = x⁴ + x² + 1, a = -2
f(a= -2) = (-2)⁴ +(-2)² +1 = 21
f'(x) = 4x³ + 2x
f'(a= -2) = 4(-2)³ + 2×(-2) = -36
f"(x) = 12x² + 2
f"( a = -2) = 12(-2)² + 2 = 50
.....
put all the values in above equation (1) we get,
f(x) = 21 - (36/2)(x+2) + (50/2!)(x + 2)² - ( 48/3!)(x+3)³ +......
f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......
c) f(x) = 2ˣ ; a = 1
f( a= 1) = 2¹ = 2
f'(x) = 2ˣ log(2)
f'(a= 1) = 2 log(2)
f"(x) = 2ˣ ( log(2) )²
f"( a = 1) = 2 ( log(2) )²
................
put all the values in above formula we get,
f(x) = 2 + 2log(2) +( 2/2!)(log2)² + (2/3!)(log2)³
+--------
f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3(log2)³(x-1)³ + --------
d) f(x) = x tan⁻¹(x²) ; a= 0
f(a=0) = 0
f'(x) = x (1/(1+x⁴) + tan⁻¹(x²)
f'(a = 0) = 0
f"(x) = ((1+x⁴) - x 4x³)/(1+x⁴)² + (1/(1+x⁴)
= (2/(1+x⁴) - 4x⁴/(1+x⁴)²
f"( a = 0) = 2 - 0 = 2
f"'(x) = - 8x³/(1+x⁴)² - 12x³/ (1+x⁴)² + 16x⁷/(1+x⁴)⁴
f"'(a = 0) = 0
..............
f(x) = 0 +0 + 2/2! (x-0)² + 0 +-------
f(x) = x² + ----
Hence, we get all required Taylor Series for
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Please help, confused which expression is equivalent???
Answer:
D
Step-by-step explanation:
m^ (-3 · -4) = m^12
n ^ (-2 · -4) = n^8
m^12 n^8
PLEASE HELP WILL MARK BARINLIEST
Answer:
d= 120
Step-by-step explanation:
c+d = 180 degree
d = 180-60 =120 degree