Answer:
Values for each variable are:
x = 19
y = -4
z = -12
Step-by-step explanation:
As we can remember the Gauss-Jordan elimination method consists of creating a matrix with all the equations of the system. Remember that, if a variable does not appear in one of the equations, we give a value of 0 to its coefficient . Each equation will constitute a line of the matrix. So, the matrix will look like this:
2 2 1 18
1 0 1 7
0 4 -3 20
For the Gauss-Jordan elimination we can multiply lines, add or subtract one line to another or we can rearrange the order at any given time. The goal is to get only 1s in the matrix diagonal, to determine the value of each variable.
Since we already have a line with a 1, we'll take that line as our starting point, and we'll rearrange it as our 1st line. By multiplying the 1st line for 2 and then subtracting the result to the second line:
1 0 1 7
0 2 -1 4
0 4 -3 20
Now, we multiply the second line by 2 and subtract the result to the third line
1 0 1 7
0 2 -1 4
0 0 -1 12
In order to get the value of Z all we have to do is multiply the third line by (-1).
1 0 1 7
0 2 -1 4
0 0 1 -12
Now, we add the third line to the second line.
1 0 1 7
0 2 0 -8
0 0 1 -12
Then, multiply the second line by a fraction 1/2, to get the value for Y
1 0 1 7
0 1 0 -4
0 0 1 -12
Finally, we subtract the third line to the 1st line to get the value for X
1 0 0 19
0 1 0 -4
0 0 1 -12
All we got left is to prove our answer is correct by replacing the variables in the system with the values found:
First equation
2(19) + 2(-4) + (-12) = 18
38 - 8 - 12 = 18
38 - 20 = 18
Second equation
19 + (-12) = 7
19 -12 = 7
Third equation
4(-4) - 3(-12) = 20
-16 + 36 = 20
A graduate weighs 35.825 g. When 10 mL of water are measured in it, the weight of the graduate and water is 45.835 g. Calculate the weight of the water and express any deviation from 10 g as a percentage of error.
Answer:
Calculated weight of water = 10.01 g
percentage error = 0.1%
Step-by-step explanation:
Given:
Weight of graduate = 35.825 g
Weight of graduate + Water = 45.835 g
Now,
The weight of water = ( Weight of graduate + Water ) - Weight of graduate
or
The weight of water = 45.835 - 35.825
or
The weight of water = 10.01 g
Now,
The percentage of error = [tex]\frac{\textup{Calculated value - Actual value}}{\textup{Actual value}}\times100[/tex]
or
The percentage error = [tex]\frac{\textup{10.01 - 10}}{\textup{10}}\times100[/tex]
or
The percentage error = 0.1%
The weight of the water is 10.010 g, and the percentage of error from the expected 10 g is 0.1%
To calculate the weight of the water and express any deviation from 10 g as a percentage of error, follow these steps:
1. Calculate the weight of the water:
The weight of the water can be determined by subtracting the weight of the empty graduate from the weight of the graduate with water.
[tex]\[ \text{Weight of water} = \text{Weight of graduate and water} - \text{Weight of empty graduate} \][/tex]
Given:
- Weight of empty graduate = 35.825 g
- Weight of graduate with water = 45.835 g
[tex]\[ \text{Weight of water} = 45.835 \, \text{g} - 35.825 \, \text{g} = 10.010 \, \text{g} \][/tex]
2. Calculate the deviation from 10 g:
[tex]\[ \text{Deviation} = \text{Weight of water} - 10 \, \text{g} \][/tex]
3. Calculate the percentage of error:
[tex]\[ \text{Percentage of error} = \left( \frac{\text{Deviation}}{10 \, \text{g}} \right) \times 100\% \][/tex]
[tex]\[ \text{Percentage of error} = \left( \frac{0.010 \, \text{g}}{10 \, \text{g}} \right) \times 100\% = 0.1\% \][/tex]
Therefore, the weight of the water is 10.010 g, and the percentage of error from the expected 10 g is 0.1%.
The point P(1, 1/6) lies on the curve y = x/(5 + x). If Q is the point (x, x/(5 + x)), use a scientific calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x.
Answer:
[tex]m(x)=\frac{5}{6(5+x)}[/tex]
Step-by-step explanation:
Slope of the secant line PQ:
P : (1, 1/6)
Q : (x, x/(5 + x))
[tex]m(x)=\frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}=\frac{x/(5 + x)-1/6}{x-1}=\frac{5(x-1)}{6(5+x)(x-1)}[/tex]
The slope of the secant line PQ is 0.
Explanation:To find the slope of the secant line PQ, we need to determine the coordinates of Q and P. Given that Q has the coordinates (x, x/(5+x)), we can substitute x=1 into the equation to find Q. So, Q is (1, 1/6).
The slope of a line passing through two points can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates of P and Q into the formula:
m = (1/6 - 1/(5+1)) / (1 - 1) = (1/6 - 1/6)/(0) = 0
Therefore, the slope of the secant line PQ is 0.
Would like assistance in understanding and solving this example on Elementary Number Theory with the steps of the solution to better understand, thanks.
a) Let a = 123 and b = 76. Find (a,b) using the Euclidean algorithm. Then find x and y such that ax+by = (a,b).
b) Show that 361x+2109y = 1000 does not have integer solutions.
Answer:
The gcd(123,76) is equal to 1. The linear combination is [tex]1=-21 \cdot 123+34 \cdot 76[/tex]
The 361x+2109y = 1000 does not have integer solutions because the gcd(2109, 361) is equal to 19 and [tex]19\not| \:1000[/tex]
Step-by-step explanation:
Point a:
The greatest common divisor (GCD) of two numbers is the largest numbers that divide them both. The Euclidean algorithm is an efficient method for computing the GCD without explicitly factoring the two integers.
These are the steps:
Let a=x, b=yGiven x,y, use the division algorithm to write x=yq + r (q = quotient and r = remainder)if r=0, stop and output y; this is the gcd of a,bif r ≠ 0, replace (x,t) by (y,r): Go to step 2The division algorithm is an algorithm in which given 2 integers n and d, it computes their quotient q and remainder r, where [tex]0\leq r<|d|[/tex]. Let's say we have to divide n (dividend) by d (divisor):
Subtract d from n repeatedly.The resulting number is known as the remainder r, and the number of times that d is subtracted is called the quotient q.To compute gcd(123,76), divide the larger number by the smaller number, using the division algorithm we have:
[tex]\frac{123}{76} = 123-76 =47[/tex]
At this point, we cannot subtract 76 again. Hence 1 is the quotient ( we subtract 76 from 123 one time) and 47 is the remainder. We can express this as a linear combination [tex]123=76 \cdot 1+47[/tex].
Using the same reasoning and the steps of the Euclidean algorithm we have:
[tex]gcd(123,76)=\\123=76 \cdot 1 +47\\76=47 \cdot 1 +29\\47=29 \cdot 1 +18\\29=18 \cdot 1 +11\\18=11 \cdot 1 +7\\11=7\cdot 1 +4\\7= 4\cdot 1+3\\4 = 3 \cdot 1 +1\\3 = 1 \cdot 3 +0[/tex]
The gcd(123,76) is equal to 1.
To represent 1 as a linear combination of the integers 123 and 76, we start with the next-to-last of the above equations and successively eliminate the remainders 3, 4, 7, 11, 18, 29, and 47.
[tex]1=4-3 \cdot 1\\1=4-(7-4 \cdot 1) \cdot 1\\1=2\cdot 4 - 7\cdot 1\\1=2\cdot(11 -7 \cdot 1) -7 \cdot 1\\1=2\cdot 11 -7 \cdot 3\\1=2\cdot 11 -(18-11\cdot 1) \cdot 3\\1=5\cdot 11-3\cdot 18\\1=5\cdot (29-18\cdot 1)-3\cdot 18\\1=5\cdot 29 -8\cdot 18\\1=5\cdot 29 -8\cdot (47-29\cdot 1)\\1=13\cdot 29 -8\cdot 47\\1=13 \cdot (76-47 \cdot 1)-8\cdot 47\\1=13 \cdot 76 -21 \cdot 47\\1=13 \cdot 76 -21 \cdot (123-76\cdot 1)\\1=-21 \cdot 123+34 \cdot 76[/tex]
Point b:
We can use this theorem:
When ax + by = c is solvable. Given integers a, b, and c with a and b not both 0, there exist integers x and y such that ax + by = c if and only if (a,b) | c
In this particular expression 361x+2109y = 1000 we need to find the gcd(2109, 361) and check if gcd(2109, 361) | 1000 is true.
[tex]2109=361\cdot 5 +304\\361 = 304 \cdot 1 +57\\304= 57\cdot 5 +19\\57=19\cdot 3 +0[/tex]
The gcd(2109, 361) is equal to 19. We can see that 19 does not divide 1000 ([tex]19\not| \:1000[/tex]), that is the reason 361x+2109y = 1000 does not have integer solutions.
Find a formula for Y(t) with Y(0)=1 and draw its graph. What is Y\infty?
a. Y'+2Y=6
b. Y'+2Y=-6
Answer:
[tex](a)\ y(t)\ =\ -2e^{-2t}+3[/tex]
[tex](b)\ y(t)\ =\ 4e^{-2t}-3[/tex]
Step-by-step explanation:
(a) Given differential equation is
Y'+2Y=6
=>(D+2)y = 6
To find the complementary function, we will write
D+2=0
=> D = -2
So, the complementary function can be given by
[tex]y_c(t)\ =\ C.e^{-2t}[/tex]
To find the particular integral, we will write
[tex]y_p(t)\ =\ \dfrac{6}{D+2}[/tex]
[tex]=\ \dfrac{6.e^{0.t}}{D+2}[/tex]
[tex]=\ \dfrac{6}{0+2}[/tex]
= 3
so, the total solution can be given by
[tex]y_(t)\ =\ C.F+P.I[/tex]
[tex]=\ C.e^{-2t}\ +\ 3[/tex]
[tex]y_(0)=C.e^{-2.0}\ +\ 3[/tex]
but according to question
1 = C +3
=> C = -2
So, the complete solution can be given by
[tex]y_(t)\ =\ -2.e^{-2.t}\ +\ 3[/tex]
(b) Given differential equation is
Y'+2Y=-6
=>(D+2)y = -6
To find the complementary function, we will write
D+2=0
=> D = -2
So, the complementary function can be given by
[tex]y_c(t)\ =\ C.e^{-2t}[/tex]
To find the particular integral, we will write
[tex]y_p(t)\ =\ \dfrac{-6}{D+2}[/tex]
[tex]=\ \dfrac{-6.e^{0.t}}{D+2}[/tex]
[tex]=\ \dfrac{-6}{0+2}[/tex]
= -3
so, the total solution can be given by
[tex]y_(t)\ =\ C.F+P.I[/tex]
[tex]=\ C.e^{-2t}\ -\ 3[/tex]
[tex]y_(0)\ =C.e^{-2.0}\ -\ 3[/tex]
but according to question
1 = C -3
=> C = 4
So, the complete solution can be given by
[tex]y_(t)\ =\ 4.e^{-2.t}\ -3[/tex]
A web site was hit 300 times over a period of 15 days Show that over some period of 3 consecutive days, it was hit at least 60 times.
Answer:
We will divide the 15 days in five periods of 3 consecutive days each.
Now to solve this we will use the pigeonhole principle.
This states that if (N+1) pigeons occupy N holes, then some hole must have at least 2 pigeons.
So, we have n=300 pigeons and k=5 holes.
[tex][\frac{n}{k} ]=[\frac{300}{5} ][/tex]
Hence, there is a period of 3 consecutive days in which the website was hit at least 60 times.
a store light flashes every 60 minutes while another light flashes open every 24 minutesif both stores begin flashing at 10 am what time will they both flash together again
Answer:
Both lights flash at 1:00
Step-by-step explanation:
they flash together every 2*7=14 minutes
in 60 minutes there are 4*14=56 minutes
they flash together at 1:56
more minutes brings you to 2:00
they flash together at 2:10 (4+10=14)
that leaves 50 minutes left in the hour
they flash three times which uses up 42 minutes
this brings you to 2:52 (10+42=52)
8 more minutes brings you to 3:00
in 6 more minutes (8+6=14) it will be 3:06 and they will flash together
Arithmetic Modular Composite Numbers (4 marks). Carry out the following calcula- tions by hand by using the Chinese Remainder Theorem to split each operation into two operations modulo smaller numbers. You must show your work to receive full credit (a) 23 x 36 mod 55 (b) 29 x 51 mod 91
Answer:
a) 23 x 36 (mod 55) = 3 (mod 55)
b) 23 x 36 (mod 55) = 23 (mod 91)
Step-by-step explanation:
The Chinese Remainder Theorem lets us split a composite modulo into its prime components and solve for smaller numbers.
a) Using the Chinese Remainder Theorem, we have that 55 = 11 x 5
Since 11 and 5 are relatively prime numbers, we can use the Theorem and rewrite 23 x 36 mod 55 as: 23 x 36 (mod 11) and 23 x 36 (mod 5).
First we will work with 23 x 36 (mod 11)
[tex](23)(36)(mod 11) = (1)(3) (mod 11)[/tex] (Since 23 is congruent to 1 modulo 11 and 36 is congruent to 3 modulo 11)
Now we do the same with 23 x 36 (mod 5)
[tex](23)(36) (mod 5) = (3)(1) (mod5) = 3 (mod 5)[/tex]
Now we will use the Chinese Remainder Theorem to solve this pair of equations:
x = 3 (mod 11) and x = 3(mod 5)
[tex]x=5y+3\\5y+3=3(mod 11)\\5y=0(mod 11)\\y=0 (mod 11)\\y=11z\\x=5(11z)+3\\x=55z + 3\\x=3(mod 55)\\[/tex]
b) We are going to use the same procedure from a)
91 = 13 x 7
29 x 51 (mod 91) = 29 x 51 (mod 13) and 29 x 51 (mod 7)
29 x 51 (mod 13) = 3 x 12 (mod 13) = 36 (mod 13) = 10 (mod 13)
29 x 51 (mod 7) = 1 x 2 (mod 7) = 2 (mod 7)
Our pair of equations is x = 10 (mod 13) and x = 2 (mod 7)
[tex]x= 7y + 2\\7y + 2 = 10(mod13)\\7y= 8(mod13)\\y= 3 (mod 13)\\y=13y+3\\x=7(13y+3) + 2\\x=91y +21+2\\x=91y+23\\x= 23 (mod 91)[/tex]
A sales representative for a Children's Fashion store earns a salary of $1100.00 per month plus a commission based on the total sales. During the month of January, this sales representative's total sales were $8300.00 and was paid $2130.00. What is this sales representative's commission rate?
a.
14.30963855%
b.
11.10963855%
c.
13.60963855%
d.
12.40963855%
e.
14.60963855%
f.
None of the above.
Answer:
Option d. 12.40963855% is the answer.
Step-by-step explanation:
Monthly salary of a sales representative = $1,100
He was paid for the month = $2,300
His total sales were = $8,300
His commission for the month of sales = Total payment - monthly salary
= 2,300 - 1,100
= $1,030
He got $1,030 as commission on the sales of $8,300.
The percentage of the commission = [tex]\frac{1030}{8300}\times 100[/tex]
= 12.40963855421687%
Option d. 12.40963855% is the answer.
Initially tank I contains 100 litres of salt brine with a concentration of 1 kilogram per litre, and tank II contains 100 litres of water. Liquid is pumped from tank I into tank II at a rate of 1 litre per minute, and liquid is pumped from tank II into tank I at a rate of 2 litres per minute. The tanks are kept well stirred. Let A1 be the amount of salt in kilograms in tank I and A2 be the amount of salt in pounds in tank II.
(a) Calculate A1(t) and C1(t). For which range of values of t are the expression for A1(t) and C1(t) valid?
(b) What is the concentration in tank I after 10 minutes?
Answer:
a)[tex]A1(t)=\frac{100000000}{(100-t)(100+t)^{2} } \\C1(t)=\frac{A1(t)}{100+t}[/tex]
b) C1 = 0.8348 [kg/lt]
Step-by-step explanation:
Explanation
First of all, the rate of change of the amount of salt in the tank I is equal to the rate of change of salt incoming less the rate change of the salt leaving, so:
[tex]\frac{dA1(t)}{dt}= R_{in}C_{in}-R_{out}C_{out}[/tex]
We know that the incoming rate is greater than the leaving rate, this means that the fluid in the tank I enters more than It comes out, so the total rate is :
[tex]R_{total}=R_{in}-R_{out}=\frac{2 lt}{min} - \frac{1 lt}{min}= \frac{1 lt}{min}[/tex]
This total rate means that 1 lt of fluid enters each minute to the tank I from the tank II, with the total rate we can calculate the volume in the tank I y tank II as:
[tex]V_{I}=100 lt + Volumen_{in}= 100 lt + (\frac{1lt}{min})(t) =100+t[/tex]
[tex]V_{II}=100 lt - Volumen_{out}= 100 lt - (\frac{1lt}{min})(t) =100-t[/tex]
Now we have the volume of both tanks, the next step is to calculate the incoming and leaving concentration. The concentration is the ratio between the amount of salt and the volume, so:
[tex]C(t)=C_{out} =\frac{A1(t)}{V_{I} }=\frac{A1(t)}{100+t }[/tex]
Since fluid is pumped from tank I into tank II, the concentration of the tank II is a function of the amount of salt of the tank I that enters into the tank II, thus:
[tex]C_{in} =\frac{(A1(t)/V_{I})(t)}{V_{II} }=\frac{A1(t)}{V_{I} V_{II}}(t)[/tex]
[tex]C_{in} =\frac{A1(t)}{(100+t)(100-t)}(t)=\frac{A1(t)}{(10000-t^{2} )}(t)[/tex]
If we substitute the concentrations and the rates into the differential equation we can get:
[tex]\frac{dA1(t)}{dt}= R_{in}C_{in}-R_{out}C_{out}\\\frac{dA1(t)}{dt}= (2)(\frac{(t)A1(t)}{10000-t^{2} })-(1)(\frac{(A1(t)}{100+t })[/tex]
[tex]\frac{dA1(t)}{dt}= A1(t)(\frac{2t}{10000-t^{2} }-\frac{1}{100+t })[/tex]
[tex]\frac{dA1(t)}{dt}- (\frac{2t}{10000-t^{2} }-\frac{1}{100+t })A1(t)=0[/tex]
The obtained equation is a homogeneous differential equation of first order and the solution is:
a) [tex]A1(t)= \frac{100000000}{(100-t)(100+t)^{2} }[/tex]
and the concentration is:
[tex]C1(t)= \frac{100000000}{(100-t)(100+t)^{3}}[/tex]
This equations A1(t) and C1(t) are only valid to 0<=t<100 because to t >=100 minutes the tank II will be empty and mathematically A1(t>=100) tends to the infinite.
b) To calculate the concentration in the tank I after 10 minutes we have to substitute t=10 in C1(t), thus:
[tex]C1(10)= \frac{100000000}{(100-10)(100+10)^{3}}=0.8348 kg/lt[/tex]
15.) A market
research worker interviewed a random sample of 18people about their
use of a certain product. The results, in termsof Yes (Y) or No (N)
are as follows:Y-N-N-Y-Y-Y-N-Y-N-Y-Y-Y-N-Y-N-Y-Y-N. Estimate the
populationproportion of users of the product.
Answer: [tex]p=0.610[/tex]
Step-by-step explanation:
Given : A market research worker interviewed a random sample of 18 people about their use of a certain product.
The results, in terms of Yes (Y) or No (N) are as follows:
Y-N-N-Y-Y-Y-N-Y-N-Y-Y-Y-N-Y-N-Y-Y-N.
The number of people said "Yes" for the product= 11
Then, the sample proportion for the users of the product =[tex]\hat{p}=\dfrac{11}{18}0.611111111111\approx0.61[/tex]
We know that the sample proportion is the best estimate for the population proportion.
Thus the point estimate for population proportion : [tex]p=\hat{p}=0.610[/tex]
At Phill's Discount Car Kingdom, you can purchase a used car for $14,000. Phill offers you two payment options: Option 1: You can apply an immediate cash rebate of $2,000 from the dealer to reduce the cost of the car and finance the rest with a loan that has an annual rate of 3.6%, with interest compounded monthly, for 3 years. You would make equal payments at the end of each month until the loan was repaid . Option 2: You can take out a 0% loan for the full price of the car in which you agree to pay the same amount at the end of each month for 3 years until the car is paid off What is the total amount that you would pay (out of pocket) for the car under each option?
Answer:
Ans. For option 1, you would pay a total of $14,677.64 and for the second option, you would pay $14,000.
Step-by-step explanation:
Hi, we need to find the amount of the equal payments that you need to make every month, given the problem´s conditions. First, let´s find the effective montly rate of this credit.
[tex]EffectiveMonthlyRate=\frac{Rate(Compounded Monthly)}{12}[/tex][tex]EffectiveMonthlyRate=\frac{0.036}{12} =0.003[/tex]
This means that the rate is 0.3% effective monthly
The period of time for this obligation is 3 years, but since the payments are made every month, we need to use 36 months instead of 3 years.
Now, we are ready to find the amount of money that you need to pay every month, for 36 months in order to pay for your car. We use the following formula.
[tex]PresentValue=\frac{A((1+r)^{n}-1) }{r(1+r)^{n} }[/tex]
Since you made a down payment of $2,000, we will only need to finance $12,000. This is the way everything should look like.
[tex]12,000=\frac{A((1+0.003)^{36}-1) }{0.003(1+0.003)^{36} }[/tex]
Let´s solve for A (annuity)
[tex]12,000=\frac{A(0.11386764 }{0.003416 }[/tex]
[tex]12,000==A(34.0757554)[/tex]}
[tex]\frac{12,000}{34.0757554} =A=352.17[/tex]
The total amount paid if you take this option is:
[tex]Amount Paid=2,000+352.17*36=14,677.64[/tex]
In the case of option 2 (0% loan-pay same amount every month for 36 months), there is no need for any calculations (because you pay $14,000 in total), but if you want to know how much to pay every month, you just go ahead and divide 14,000 by 36 which is $388.89. But at the end, this way you will pay $14,000.
Best of luck.
A cell phone company has a fixed cost of $1,000,000 per month and a variable cost of $20 per month per subscriber. The company charges $29.95 per month to its cell phone customers. (2) a. What is the breakeven point for this company?
Answer:
100503
Step-by-step explanation:
Data provided in the question:
Fixed cost per month for the cell phone company = $1,000,000
Variable cost per month per subscriber = $20
Charges for the customer per month = $29.95
Now,
the breakeven point is calculated as:
Breakeven point = [tex]\frac{\textup{Total fixed cost}}{\textup{Charges - variabel cost}}[/tex]
on substituting the respective values, we get
Breakeven point = [tex]\frac{\textup{1,000,000}}{\textup{29.95 - 20}}[/tex]
or
Breakeven point = 100502.51 ≈ 100503
3. Galshore of Maloat is making a journey from the island of Gont to Oranea. The distance from the two lands is 965 miles. Galshore travels in her Sea Speeder at 129 miles per hour. How long did it take Galshore to make her journey? 4. The Countess Aliskia travels from Carthmore to Glenwyr. The distance between the two places is 612.1 miles. She makes the trip in Via her golden Sky Chariot. Via, with his emerald eyes can be seen travelling at a speed of 324 miles per hour with an altitude of 3 miles above the surface of the earth. How long did it take for the Countess to complete her magnificent trip?
Answer:
3. about 7.48 hours
4. about 1.8892 hours
Step-by-step explanation:
Both of these questions make use of the relation ...
time = distance / speed
3. time = (965 mi)/(129 mi/h) ≈ 7.48062 h ≈ 7.48 h
__
4. time = (612.1 mi)/(324 mi/h) ≈ 1.889198 h ≈ 1.8892 h
(We have rounded the time to sufficient precision so that the distance at the given speed rounds to the number given.)
Suppose you invest $750.00 in a fund earning 6% simple discount. A certain time later you withdraw the investment (principal and interest) and invest it in another fund earning 3.5% compound interest for two years. How much total time (INCLUDING THE TWO YEARS earning compound interest) will be required for the original $750.00 to accumulate to $1,000.00? (two decimal places)
Answer:
4.08 + 2 = 6.08 years
Step-by-step explanation:
we know that
Simple Interest(S.I.) = (P × R × T) ÷ 100
where, P = Principal = 750
R = Rate = 6%
T = unknown
⇒ S.I. = (750 × 6 × t)÷ 100
⇒ S.I. = 45t
Also, Amount = S.I + Principal
⇒ Amount = 750 + 45t
Now Formula for Compound Interest is:
[tex]A = P(1+\frac{r}{100})^{t}[/tex]
where A = Amount
=1000
P = Principle
r = rate
t = total number of year
Here, P = 750 + 45t, r = 3.5% , and t = 2.
Putting all these values in above formula:
[tex]1000 = (750 + 45t)(1+\frac{3.5}{100})^{2}[/tex]
⇒ [tex]1000 = (750 + 45t)(1.071)[/tex]
⇒ t = 4.08
Hence, total time required will be 2 + 4.08 = 6.08 years.
A town's population is currently 30,000. If the population doubles every 45 years, what will the population be 90 years from now?
A. 240,000 people
B. 150,000 people
C. 120,000 people
D. 60,000 people
Answer:
C. 120,000 people
Step-by-step explanation:
First thing I did was divide 90 by 45 which gave me 2 as my answer. So now I know that the population 90 years from now will be doubled twice (x4). So I did...
[tex]30,000 \times 4 = 120,000[/tex]
Answer:
80,000 people
Step-by-step explanation:
First, find out how many times the population will double. Divide the number of years by how long it takes for the population to double.
68÷34=2
The population will double 2 times.
Now figure out what the population will be after it doubles 2 times. Multiply the population by 2 a total of 2 times.
20,00022=80,000
That calculation could also be written with exponents:
20,00022=80,000
After 68 years, the population will be 80,000 people.
Mathematics with applications in the management, Natural, and Social Sciences Twelfth edition
Chapter 6: Systems of Linear Equations and Matrices
6.1 Exercises
23.) According to Google Trends, popular interest in LED lightbulbs has been soaring. while interest in CFLs has been dropping. The following equations approximate the Google Trends rating (on a scale from 0-100) in year x, where x = 10 corresponds to the year 2010. ( Data from www.google.com/trends.)
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
When did interest in ED lighting Surpass CFL lighting? Round your answer to the nearest year.
Answer: 2014
Step-by-step explanation:
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
we need to find which year it was the same to know where LED lighting surpassed CFL lighting
-25x + 6y = 20
15x + 2y = 322 (-3)
-25x + 6y = 20
-45x - 6y = -966
-70x = -946
x = 13.51
The nearest year would be 14 which is 2014
By solving the system of equations, we find that the interest in ED lighting surpassed CFL lighting around the year 2014.
Explanation:To find out when the interest in ED lighting surpassed CFL lighting, we'll need to solve the system of equations given by:
LED: -25x + 6y = 20
CFL: 15x + 2y = 322
We can use substitution or elimination methods to solve this system of equations. For the elimination method, multiply the first equation by 2 and the second equation by 6 to make the y-coefficients equal:
-50x + 12y = 40
90x + 12y = 1932
Now, subtract the first equation from the second:
140x = 1892. Hence, x = 1892/140 = 13.51
So, the interest in ED lighting surpassed CFL lighting in the year 2013.51. Since we round to the nearest year, we can say that this happened around the year 2014.
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If n is an odd integer, then 5n+ 3 is even.
Construct a know-show table for this statement.
Answer:
If n is an odd integer, then 5n +3 is even.
Step-by-step explanation:
Below is the know-show table for the statement
Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z=cos(x^2+y^2) that lies inside the cylinder x^2+y^2=1
If we substitute [tex]x=r\cos\theta[/tex] and [tex]y=r\sin\theta[/tex], we get [tex]r^2=x^2+y^2[/tex], so that
[tex]z=\cos(x^2+y^2)=\cos(r^2)[/tex]
which is independent of [tex]\theta[/tex], which in turn means the surface can be treated like a surface of revolution.
Consider the function [tex]f(t)=\cos(t^2)[/tex] defined over [tex]0\le t\le1[/tex]. Revolve the curve [tex]C[/tex] described by [tex]f(t)[/tex] about the line [tex]t=0[/tex]. The area of the surface obtained in this way is then
[tex]\displaystyle2\pi\int_C\mathrm dS=2\pi\int_0^1\sqrt{1+f'(t)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle2\pi\int_0^1\sqrt{1+(-2t\sin(t^2))^2}\,\mathrm dt[/tex]
[tex]=\displaystyle2\pi\int_0^1\sqrt{1+4t^2\sin^2(t^2)}\,\mathrm dt\approx7.4144[/tex]
The problem is solved by converting the variables in the surface equation into polar coordinates and expressing the surface area as a double integral which is estimated by numerical techniques to give a result of A = 4.5 m².
Explanation:The question is asking to find the surface area of the part of the surface z=cos(x^2+y^2) that lies within the cylinder x^2+y^2=1. To solve this, we must convert the variables into polar coordinates. Using the fact that x^2+y^2=r^2 in polar coordinates, our z function transforms into z = cos(r^2). Our area element in polar coordinates is r * dr * dθ. Plug these values into the surface area formula:
Surface Area = ∫∫ sqrt[1 + ((∂z/∂r)^2 +(∂z/∂θ)^2)] * r dr dθ.
Now it’s a matter of calculating the derivatives, plugging them into the formula and performing the double integral. Unfortunately, in this case, it is not straightforward to calculate the integral analytically so we use numerical techniques (like Riemann sum, Simpson rule, etc.) or a calculator with these built-in functions to estimate this integral, which gives the result A = 4.5 m² accurate up to 2 significant figures.
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Find the area of the region described.
The region between the line y=x and the curve y=2x√(25 - x^2) in the first quadrant. The total area of the shaded region is ___ (from 7087 to 100).
Answer:
The area is [tex]\frac{567}{8}u^2[/tex]
Step-by-step explanation:
The area of a flat region bounded by the graphs of two functions f (x) and g (x), with f (x)> g (x) can be found through the integral:
[tex]\int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
The integration limits are given by the intersection points of the graphs of the functions in the first quadrant. Then, the cut points are:
[tex]g(x) = x\\f(x) = 2x\sqrt{25-x^2}[/tex]
[tex]x=2x\sqrt{25-x^2}\\x^2=4x^2(25-x^2)\\x^2(1-100+4x^2)=0\\x_1=0\\x_2=\frac{3\sqrt{11}}{2}[/tex]
The area of the region is:
[tex]\int\limits^b_a {[f(x) - g(x)]} \, dx = \int\limits^{\frac{3\sqrt{11}}{2}}_0 {x(2\sqrt{25-x^2}-1)} \, dx = \frac{567}{8}u^2[/tex]
Which correlation coefficient corresponds to the best-fit line that most closely models it’s set of data. A. -0.87, B. -0.15. C. 0.13. D. 0.84
Answer:
A. -0.87
Step-by-step explanation:
The correlation coefficient is a measure of the strength and nature of the linear relationship between two variables, one dependent and one independent. This coefficient takes values between -1 and 1, indicating with its sign whether the relationship is direct or inverse between the variables involved and with its absolute value indicates the strength of the linear relationship between them. A coefficient with absolute value close to 1 indicates great strength and better fit.
Conclusion: The best coefficient, of the propuetso in the problem, is that of -0.87, which indicates a strong relationship between the variables, a good fit and an inverse relationship between them.
A company estimates that the marginal cost (in dollars per item) of producing x items is 1.65 − 0.002x. If the cost of producing one item is $570, find the cost of producing 100 items. (Round your answer to two decimal places.)
Answer:
The cost of producing 100 items is $723.35
Step-by-step explanation:
The marginal cost is the derivative of the total cost function, so we have
[tex]C^{'}(x)=1.65-0.002x[/tex]
To find the total cost function we need to do integration
[tex]C(x)= \int\, C^{'}(x)dx \\C(x)=\int\,(1.65-0.002x) dx[/tex]
Apply the sum rule to find the integral
[tex]\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)[/tex]
[tex]\int \:1.65dx=1.65x\\\int \:0.002xdx=0.001x^2[/tex]
[tex]C(x)=\int\,(1.65-0.002x) dx = 1.65x-0.001x^2+D [/tex]
D is the constant of integration
We are given that C(1) = $570, we can use this to find the value of the constant in the total cost function
[tex]C(1)=570=1.65*(1)+0.001*(1)^2+D\\D=570-1.649=568.351[/tex]
So the total cost function is [tex]C(x)=1.65x-0.001x^2+568.351 [/tex] and the cost of producing 100 items is
x=100
[tex]C(100)=1.65*(100)-0.001*(100)^2+568.351 = 723.35[/tex]
A telemarketer makes a sale on 25% of his calls. If he makes 300 calls in a night, what is the probability that he will make more than 70 sales but less than 90 sales?
Answer: 0.7258
Step-by-step explanation:
Given : A telemarketer makes a sale on 25% of his calls.
i.e. p=0.25
He makes 300 calls in a night, i.e. n=300
Let x be a random variable that represents the number of calls make in night.
To convert the given binomial distribution to normal distribution we have :-
[tex]\mu=np=300(0.25)=75[/tex]
[tex]\sigma=\sqrt{p(1-p)n}=\sqrt{(0.25)(1-0.25)(300)}\\\\=\sqrt{56.25}=7.5[/tex]
Now, using [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-value corresponds to x= 70 :-
[tex]z=\dfrac{70-75}{7.5}\approx-0.67[/tex]
The z-value corresponds to x= 90 :-
[tex]z=\dfrac{90-75}{7.5}\approx2[/tex]
By using the standard normal distribution table for z, the probability that he will make more than 70 sales but less than 90 sales:-
[tex]P(-0.67<z<2)=P(z<2)-P(z<-0.67)\\\\=P(z<2)-(1-P(z<0.67))\\\\=0.9772-(1-0.7486)\\\\=0.9772-0.2514=0.7258[/tex]
Hence, the probability that he will make more than 70 sales but less than 90 sales= 0.7258
How does logical operator work- explain the differences between A AND B and A OR B? (6 pts)
Answer with Step-by-step explanation:
We are given two input A and B
A AND B=[tex]A\cdot B[/tex]
If A=0 and B=0 then [tex]A\cdot B=0[/tex]
If A=0 and B=1 then [tex]A\cdot B=0[/tex]
If A =1 and B=0 then [tex]A\cdot B=0[/tex]
If A=1 and B=1 then [tex]A\cdot B=1[/tex]
A OR B=A+B
If A=0 and B=0 then A+B=0+0=0
If A=0 and B=1 then A+B=0+1=1
If A =1 and B=0 then A+B=1+0=1
If A=1 and B=1 then A+B=1+1=1
If A=0 and B=1 or A=1 and B=0 then A AND B=0 but A OR B=1
This is the main difference A AND B and A OR B.
Norgestrel and ethinyl estradiol tablets are available containing 0.5 mg of norgestrel and 50 μg of ethinyl estradiol. How many grams of each ingredient would be used in making 10,000 tablets?
Answer:
5g of norgestel are used in making 10,000 tablets.
0.5g of ethinyl estradiol are used in making 10,000 tablets.
Step-by-step explanation:
This problem can be solved as a rule of three problem.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
First step: Grams of norgestrel
Each tablet contais 0.5mg of norgestrel. How many miligrams are in 10,000 tablets?
1 tablet - 0.5 mg
10,000 tablets - x mg
x = 10,000*0.5
x = 5,000 mg
Now we have to convert 5,000 mg to g. Each g has 1,000 mg. So:
1g - 1,000 mg
xg - 5,000 mg
1,000x = 5,000
[tex]x = \frac{5,000}{1,000}[/tex]
x = 5g,
5g of norgestel are used in making 10,000 tablets.
Final step: Grams of ethinyl estradiol
50ug = 0.05 mg.
So
1 tablet - 0.05mg
10,000 tablets - xg
x = 10,000*0.05
x = 500 g
Now we have to convert 500 mg to g. Each g has 1,000 mg. So:
1g - 1,000 mg
xg - 500 mg
1,000x = 500
[tex]x = \frac{500}{1,000}[/tex]
x = 0.5g,
0.5g of ethinyl estradiol are used in making 10,000 tablets.
There are 129 schools in the NCAA's Division 1 Football Bowl Subdivision. Of these, 115 have nicknames that end in "s" (like the UCLA Bruins) 19 have nicknames that involve a color (the Stanford Cardinal), and · 13 nicknames involve both a color and end in "s" (the California Golden Bears). How many teams have nicknames without a color and don't end in "s?
Answer:
There are 8 teams that have nicknames without a color and don't end in "s.
Step-by-step explanation:
This can be solved by treating each value as a set, and building the Venn Diagram of this.
-I am going to say that set A are the teams that have nicknames that end in S.
-Set B are those whose nicknames involve a color.
-Set C are those who have nicknames without a color and don't end in "s.
We have that:
[tex]A = a + (A \cap B)[/tex]
In which a are those that have nickname ending in "s", but no color, and [tex]A \cap B[/tex] are those whose nickname involves a color and and in "s".
By the same logic, we have
[tex]B = b + (A \cap B)[/tex]
In which b are those that nicknames involves a color but does not end in s.
We have the following subsets:
[tex]a,b, (A \cap B), C[/tex]
There are 129 schools, so:
[tex]a + b + (A \cap B) + C = 129[/tex]
Lets find the values, starting from the intersection.
The problem states that:
13 nicknames involve both a color and end in "s". So:
[tex]A \cap B = 13[/tex]
19 have nicknames that involve a color. So:
[tex]B = 19[/tex]
[tex]B = b + (A \cap B)[/tex]
[tex]b + 13 = 19[/tex]
[tex]b = 6[/tex]
115 have nicknames that end in "s". So:
[tex]A = 115[/tex]
[tex]A = a + (A \cap B)[/tex]
[tex]a + 13 = 115[/tex]
[tex]a = 102[/tex]
Now, we just have to find the value of C, in the following equation:
[tex]a + b + (A \cap B) + C = 129[/tex]
[tex]102 + 6 + 13 + C = 129[/tex]
[tex]C = 129 - 121[/tex]
[tex]C = 8[/tex]
There are 8 teams that have nicknames without a color and don't end in "s.
(x-5)^2=81, solve for x
Answer:
x ∈ {-4, 14}
Step-by-step explanation:
Take the square root and add 5.
(x -5) = ±√81 = ±9
x = 5 ± 9
x ∈ {-4, 14} . . . . . . x may be either of -4 or 14
Tim has just inherited £1000 and he decides not to invest it but rather to spend 10% of the remaining money each month. Each month Tim's saving thus reduce by a factor of 0.9 Calculate, to the nearest integer, the amount remaining (in E) after 20 months. You may find it useful to first devise a formula for the amount remaining after n months. Enter your answer, to the nearest integer and without units, in the box below. Answer:
Answer:
£ 121.57
Step-by-step explanation:
As given in question,
Total amount inherited by Tim = £1000
He spends each month = 10 % of remaining money
savings reduced each month = 0.9 of remaining amount
So, the amount of money after one month = 0.9 x £1000
amount of money remained after 2 months = 0.9 x 0.9 x £1000
amount of money remained after 3 months = 0.9 x 0.9 x 0.9 x £1000
Hence, the amount of money remained after the n months can be given by,
[tex]E\ =\ 0.9^n\times 1000[/tex]
Hence, amount of money remained after 20 months can be given by,
[tex]E_{20}\ =\ 0.9^{20}\times1000[/tex]
= £ 121.57
So, the amount of money remained after 20 months will be £ 121.57.
a plant grows at a rate of 7 ft every two weeks
Answer:
0 is 0,1 is 7,2 is 14 and 3 is 20 (c. is that it has not grown yet and (d. is 7 per week
Answer:
Step-by-step explanation:
A. (0,0) (1,7) (2,14) (3,21)
C. (0,0) is the starting point
Show that any integer n > 12 can be written as a sum 4r + 5s for some nonnegative integers r, s. (This problem is sometimes called a postage stamp problem. It says that any postage greater than 11 cents can be formed using 4 cent and 5 cent stamps.)
Answer:
Use induction for the prove
Step-by-step explanation:
Mathemathical induction is an useful method to prove things over natural numbers, you check for the first case, supose for the n and prove using your hypothesis for n+1
there says any integer bigger than 12 can be written as 4r+5s
so first number n can be is 13.
we can check n=13 = 4*2+5*1 r=2 and s=1 give 13.
Now we suppose n can be written as 4r+5s
and we can check if n+1=4r'+5s' with r' and s' integers.
we replace n as 4r+5s because that is our hypotesis
n+1=4r+5s+1
if we write that 1 as 5-4
4r+5s+1
4r+5s+5-4
then we can write
4(r-1)+5(s+1) , we got n+1= 4 (r-1) +5(s+1) where r-1 and s+1 are non negative integers. because r and s were no negative integers ( if r is not 0)
what if r=0?
if r is 0 , n is a multiple of 5 and n+1 can be written as 5s+1
first multiple of 5 we can write is 15 since n is bigger than 12 , then smaller s is 3.
for any n+1 we can write
n+1=5s+1=5 (s-3) +3*5+1=5(s-3)+4*4, s-3 is 0 or bigger.
(check 3*5+1 is 16, the same as 4*4)
Final answer:
Any integer greater than 12 can be expressed as the sum 4r + 5s with nonnegative integers r and s by providing concrete examples for n values from 13 to 17 and then proving for any n > 17 using number theory.
Explanation:
To show that any integer n greater than 12 can be written as the sum 4r + 5s for some nonnegative integers r and s, we can provide examples and create a general proof. We know that for n = 13 through n = 17, specific values of r and s can be found that satisfy the equation:
n = 13 = 4(1) + 5(1)
n = 14 = 4(3) + 5(0)
n = 15 = 4(0) + 5(3)
n = 16 = 4(4) + 5(0)
n = 17 = 4(1) + 5(3)
For n > 17, we can write n = 17 + k, where k is a nonnegative integer. Since every nonnegative integer can be expressed as a multiple of 4 plus an additional 0, 1, 2, or 3 (as k = 4q + r, with 0 ≤ r < 4), we can substitute into our equation for n to n = 17 + 4q + r. The key is to notice that for any additional 4 we add to the sum, we can simply increase r (our existing count of 4-cent stamps), ensuring that the equation 4r + 5s will always hold.
This proof shows that we can always add enough 4-cent stamps (or subtract them and add a 5-cent stamp) to reach any postage amount above 12 cents. This is a classic example of a problem that uses the concept of number theory and diophantine equations.
[(√4 + 3)2 – 9] / (√9 – 1)3 * 2
Answer: 1
Step-by-step explanation:
[(√4 + 3)² – 9] / (√9 – 1)³ * 2 =
[(2 + 3)² – 9] / (3 – 1)³ * 2 =
[(5)² – 9] / (2)³ * 2 =
[25 – 9] / 8 * 2 =
[16] / 16 = 1