Answer:
[tex][(1+i\sqrt{29}), (1-i\sqrt{29})][/tex], [tex][(1+i\sqrt{26}), (1-i\sqrt{26})][/tex], [tex][(1+i\sqrt{21}), (1-i\sqrt{21})][/tex],
[tex][(1+i\sqrt{14}), (1-i\sqrt{14})][/tex], and [tex][(1+i\sqrt{5}), (1-i\sqrt{5})][/tex].
Step-by-step explanation:
Given that two complex numbers have a product of 30 and the real part of the complex numbers are positive integers.
Let one complex number is [tex]a+ib[/tex] where [tex]a[/tex] is a positive integer and [tex]b[/tex] is a real number.
As the product of two complex numbers is a purely real number, so the other complex number must be the conjugate of the first complex number.
So, another complex number [tex]= a-ib[/tex]
As a product of both the numbers = 30
[tex]\Rightarrow (a+ib)(a-ib)=30 \\\\\Rightarroe a^2 - (ib)^2=30 \\\\\Rightarrow a^2 - (i^2b^2)=30 \\\\Rightarrow a^2 - (-1\times b^2)= 30[/tex] [as i^2=-1]
[tex]\Rightarrow a^2+b^2= 30 \\\\\Rightarrow b^2 = 30-a^2 \\\\[/tex]
[tex]\Rightarrow b=\pm \sqrt {30-a^2} \cdots(i)[/tex]
Now, as a is a positive integer, so by taking possible values of a, we can determine the values of b from the equation (i).
For [tex]a=1[/tex]
[tex]b=\pm \sqrt {30-1^2}=\pm \sqrt {29}[/tex]
The positive sign is for the one number and the negative sign is for the conjugate number (another number).
So, the 1st combination is [tex][(1+i\sqrt{29}), (1-i\sqrt{29})][/tex]
Similarly,
For [tex]a=2[/tex]
[tex]b=\pm \sqrt {30-2^2}=\pm \sqrt {26}[/tex]
So, the 2nd combination is [tex][(1+i\sqrt{26}), (1-i\sqrt{26})][/tex]
For [tex]a=3[/tex]
[tex]b=\pm \sqrt {30-3^2}=\pm \sqrt {21}[/tex]
So, the 3rd combination is [tex][(1+i\sqrt{21}), (1-i\sqrt{21})][/tex]
For [tex]a=4[/tex]
[tex]b=\pm \sqrt {30-4^2}=\pm \sqrt {14}[/tex]
So, the 4th combination is [tex][(1+i\sqrt{14}), (1-i\sqrt{14})][/tex]
For [tex]a=5[/tex]
[tex]b=\pm \sqrt {30-5^2}=\pm \sqrt {5}[/tex]
So, the 5th combination is [tex][(1+i\sqrt{5}), (1-i\sqrt{5})][/tex]
The higher integral value of [tex]a[/tex] is not possible for any real [tex]b[/tex].
Hence, the possible combinations of required numbers are
[tex][(1+i\sqrt{29}), (1-i\sqrt{29})][/tex], [tex][(1+i\sqrt{26}), (1-i\sqrt{26})][/tex], [tex][(1+i\sqrt{21}), (1-i\sqrt{21})][/tex],
[tex][(1+i\sqrt{14}), (1-i\sqrt{14})][/tex], and [tex][(1+i\sqrt{5}), (1-i\sqrt{5})][/tex].
The concentration of dichlorodiphenyltrichloroethane (DDT - an infamous pesticide) in Lake Michigan has been declining exponentially over many years. In 1971, the DDT concentration was 13.00 ppm (parts per million). In 1984, the DDT concentration was 2.22 ppm.
Using these data points, determine the exponential decay function to model the DDT concentration decline in terms of years, tt, where t=0t=0 represents the year 1971.
A) Write your function as y = Ce^{rt}y=Ce^rt. Round your value of rr to four decimal places.
B) Another way to write the function down is known as the half-life formula, written as y = C(0.5)^{t/H}y=C(0.5)^t/H, where HH is the time required for the concentration to get cut in half. Using the initial two data points, find the value of HH, rounded to 2 decimal places. Show all Algebraic work.
C) The goal was to reduce the DDT concentration to 0.98 ppm by 1995. Was this possible according to your model?
Answer: A) [tex]y=13e^{-0.1359t}[/tex]
B) H = 5.10
C) Yes
Step-by-step explanation: Exponential Decay function is a model that describes the reducing of an amount by a constant rate over time. Generally, it is written in the form: [tex]y(t)=Ce^{rt}[/tex]
A) C is initial quantity, in this case, the initial concentration of DDT. To determine r, using the data given:
[tex]y(t)=Ce^{rt}[/tex]
[tex]2.22=13e^{13r}[/tex]
[tex]e^{13r}=0.1708[/tex]
Using a natural logarithm property called power rule:
[tex]13r=ln(0.1708)[/tex]
[tex]r=\frac{ln(0.1708)}{13}[/tex]
[tex]r=-0.1359[/tex]
The decay function for concentration of DDT through the years is [tex]y(t)=13e^{-0.1359t}[/tex]
B) The value of H is calculated by [tex]y=C(0.5)^{\frac{t}{H} }[/tex]
[tex]2.22=13(0.5)^{\frac{13}{H} }[/tex]
[tex](0.5)^{\frac{13}{H} }=0.1708[/tex]
Again, using power rule for logarithm:
[tex]\frac{13}{H} log(0.5)=log(0.1708)[/tex]
[tex]\frac{13}{H} =\frac{log(0.1708)}{log(0.5)}[/tex]
[tex]\frac{13}{H} =2.55[/tex]
H = 5.10
Constant H in the half-life formula is H=5.10
C) Using model [tex]y(t)=13e^{-0.1359t}[/tex] to determine concentration of DDT in 1995:
[tex]y(24)=13e^{-0.1359.24}[/tex]
y(24) = 0.5
By 1995, the concentration of DDT is 0.5 ppm, so using this model is possible to reduce such amount and more of DDT.
A population of beetles are growing according to a linear growth model. The initial population (week 0) is P0=9, and the population after 7 weeks is P7=37.
Find an explicit formula for the beetle population after n weeks.
After how many weeks will the beetle population reach 93?
Answer:
y = 4x + 9
week 21
Step-by-step explanation:
first I did 37-9 and got 28 I divide that by 7 ( seven weeks) and got 4 so I know the slope is 4.
so to find how many weeks till 93 I did; 93 = 4x + 9. We have to get x by its self so I minus 9 by both sides and got 84 = 4x. So now we divid both sides by 4 to get x by its self and we end up with 21 = x
A robot can complete 6 tasks in 7/10 hour. Each task takes the same amount of time. a. How long does it take the robot to complete one task? b. How many tasks can the robot complete in one hour?
Answer:
7/60 hour or 7 minutes b) 8 or 8 4/7 depending if you consider a fraction of a task is yes or no
Step-by-step explanation:
can somebody please help me omg i’m so dumb
Answer:
A
Step-by-step explanation:
what is the APPROXIMATE area of a circle with a diameter of 6.2
A.9.7 square inches
B.19.5 square inches
C.30.2 square inches
D.120.8 square inches
Answer:
k-2654 + %%12 a4 Q
Step-by-step explanation:
2323-hnbcn
e3+*
Kuel+12
78278663=h*3
james cut three square lasagnas so that each piece was 1/5 of lasagna how many pieces did James cut the lasagnas into
Answer:
The answer is 15. Each one is 5 pieces and there are 3 of them. so 5 x 3 = 15
Step-by-step explanation:
Answer:
15 pieces
Step-by-step explanation:
3*5= 15
Can someone help me please on #6 and #8 please.
Answer: yo if you look up math equation calculator and type that equation in it would solve the problem step by stem.
Step-by-step explanation: i hope this helps :D
Answer:
Number 6 is 3, Iḿ pretty sure if not sorry I tried my best
Step-by-step explanation:
To find the quotient 2/5 divided 1/4,
A. Multiply 5/2 by 4
B. Multiply 2/5 by 1/4
C. Multiply 5/2 by 4
D. Multiply 5/2 by 4
HELP ME PLEASEEEE 20 POINTS AND BRAINLY HAAAALP
Answer:
None of the choices are correct.
Step-by-step explanation:
To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.
2/5 divided by 1/4 = 2/5 times 4/1 = 2/5 times 4
Correct answer: Multiply 2/5 by 4
Answer: None of the answers are correct.
Answer:
None
Step-by-step explanation:
Write 23 between two consecutive multiples of 8. tell me the truth
Find the y-intercept of −2x + 4y = 8.
Answer: (0,2)
The y-int is 2
Answer:
the y-intercept is 2
How do you do these two questions?
Answer:
T = -22/r + 39
(a) y = [130 − 130e^(-0.015t)] / 3
(b) 22.9
Step-by-step explanation:
d²T/dr² + 2/r dT/dr = 0
If S = dT/dr, then:
dS/dr + 2/r S = 0
dS/dr = -2/r S
dS / S = -2/r dr
ln S = -2 ln r + C
S = e^(-2 ln r + C)
S = Ce^(-2 ln r)
S = C [e^(ln r)]^-2
S = C / r²
dT/dr = C / r²
dT = C dr / r²
T = -C / r + D
When r=1, T=17.
17 = -C + D
When r=2, T=28.
28 = -C/2 + D
Solving:
11 = C/2
C = 22
D = 39
T = -22/r + 39
We can check our answer by finding dT/dr and d²T/dr².
dT/dr = 22/r²
d²T/dr² = -44/r³
Plug into the original equation:
d²T/dr² + 2/r dT/dr = 0
-44/r³ + 2/r (22/r²) = 0
0 = 0
The total flow into the tank is 15 L/min, and the total flow out is 15 L/min, so the volume is constant at 1000 L. The amount of salt added to the tank is:
(0.05 kg/L) (5 L/min) + (0.04 kg/L) (10 L/min) = 0.65 kg/min
If y is the amount of salt in the tank at time t, and y/1000 is the concentration in the tank at time t, then the amount of salt removed from the tank is:
(y/1000 kg/L) (15 L/min) = 0.015y kg/min
The change in salt over time equals the amount of salt in minus the amount of salt out.
dy/dt = 0.65 − 0.015y
-200 dy/dt = 3y − 130
dy / (3y − 130) = -0.005 dt
3 dy / (3y − 130) = -0.015 dt
ln(3y − 130) = -0.015t + C
3y − 130 = Ce^(-0.015t)
3y = 130 + Ce^(-0.015t)
y = [130 + Ce^(-0.015t)] / 3
At t=0, y=0:
0 = (130 + C) / 3
C = -130
y = [130 − 130e^(-0.015t)] / 3
At t=50:
y = [130 − 130e^(-0.75)] / 3
y ≈ 22.9
2. Choose the correct answer for the following equation: *
(2x105) + (7x104) + (6x103) + (5x101)
O 27,065
O 2,765
O 27,650
O 270,650
Plssss help
Answer:
270,650 is the answer
Step-by-step explanation:
Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.)
Complete question is;
Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola. (Round your answers to the nearest hundredth.) y = 6 - x²
Answer:
height = 4 and base = 2√2
Step-by-step explanation:
Area of a rectangle is given by;
A = base(b) × height(h)
To start off, we will first have to consider half of the rectangle.
Due to the fact that it is bounded by a parabola which is symmetric over the y-axis, it means that the rectangle will have the same area on the right and left hand sides.
So if we maximize the area of the first quadrant, it means that we are also maximizing the area of the entire rectangle.
Now, since we are dealing with the base and other 2 vertices on the x-axis, it means that the base(b) of this half rectangle will be x.
The height of this rectangle is the y coordinate of the corner that is directly above the x coordinate, which is also on the rectangle.
Therefore, the coordinates of the upper corner will be (x, y), or (x, 6 - x²). Thus our half base is x and full height is 6 - x². Area will be;
A = x(6 - x²)
A = 6x - x³
We will maximize this area by finding the derivative and equating to zero.
Thus;
A' = 6 - 3x²
At A' = 0,we have;
6 - 3x² = 0
3x² = 6
x² = 6/3
x² = 2
x = √2
Thus, from y = 6 - x², we have;
y = 6 - (√2)²
y = 6 - 2
y = 4
Since x is half base, it means our full width is 2x = 2 × √2 = 2√2
So, height = 4 and base = 2√2
A woman weighed 160 pounds before dieting. After dieting she weighed 120 pounds. What percent of her weight did she lose?
160-120= 40
She lost 40 pounds
25% of 160 is 40
Percentage=160 x 25/100 =40
So she lost %25 of her weight (good job lol)
On his lunch break, Calvin purchases a burger and a drink. The total order is $8.50, of which $1.08 is tax. How much was Calvin's meal before tax?
Equation:
Answer:
Calvin's meal before tax = $7.42
Step-by-step explanation[tex] \bf : [/tex]
Cost of total order after tax = $8.50
Tax = $1.08
Cost of total order before tax = Cost of total order after tax - Tax
= $8.50 - $1.08
= $7.42
A random sample of hybrid vehicle fuel consumptions has a sample mean of x¯=53.2 mpg and sample standard deviation of s=4.8 mpg. Use the Empirical Rule to estimate the percentage of hybrid vehicle fuel consumptions that are less than 43.6 mpg. Round your answer to the nearest tenth.
Answer:
2.5%
Step-by-step explanation:
Since the mean is 53.2, a fuel consumption of 43.6 mpg is 53.2−43.6=9.6 mpg less than the mean, which is 2 standard deviations.
By the Empirical Rule, we know that about 95% of the data lies within 2 standard deviations of the mean. Therefore, 100%−95%=5% of the data lie more than 2 standard deviations away from the mean.
Since the distribution of hybrid vehicle fuel consumptions is symmetric, about half of the remaining 5% will lie to either extreme. So about 2.5% of fuel consumptions are less than 43.6 mpg.
The normal distribution is a symmetric probability distribution about the mean. The percentage of hybrid vehicle fuel consumption that is less than 43.6 mpg is 2.5%.
What is Normal Distribution?The normal distribution, also known as the Gaussian distribution, is a symmetric probability distribution about the mean, indicating that data near the mean occur more frequently than data distant from the mean. The normal distribution will show as a bell curve on a graph.
Given that the mean is 53.2, while the standard deviation is 4.8 mpg. Therefore, Accoding to the emperical rule, we can write the values as,
μ+3σ = 67.6
μ+2σ = 62.8
μ+σ = 58
μ = 53.2
μ-σ = 48.4
μ-2σ = 43.6
μ-3σ = 38.8
Therefore, The percentage of hybrid vehicle fuel consumption that is less than 43.6 mpg is,
Percentage = (0.15 + 2.35)% = 2.5%
Hence, The percentage of hybrid vehicle fuel consumption that is less than 43.6 mpg is 2.5%.
Learn more about Normal Distribution:
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CAN YOU PLZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ HELP (40 points)
The perimeter of a rectangle is 54 inches. Its length is 22 inches. Write and solve an equation to determine its width?
2 times the width + 2 times the length = perimeter
Answer:15 cm2
Step-by-step explanation:
A flying disc in the shape of a circle has a circumference of 8 inches. What is the radius of the flying disc.
Answer:
A.
Step-by-step explanation:
Recall that circumference is given by:
[tex]C=2\pi r[/tex]
We know that the circumference is 8π inches. Thus, we can substitute 8π for C:
[tex]8\pi =2\pi r[/tex]
To solve for r, the radius, we can divide both sides by 2π. Hence:
[tex]\displaystyle r = \frac{8\pi}{2\pi}=4[/tex]
Therefore, the radius is four inches.
Our answer is A.
Find the value of n. 0.09 × n = 0.054
HELP PLEASEEEEEEEEEEEEEEEEEEE
Answer:
n=0.6
Step-by-step explanation:
We want to get n alone so we can divide each side by 0.09.
So then n= 0.054/0.09.
n=0.6
Hope this helps! If you have any questions on how I got my answer feel free to ask. Stay safe!
A recipe says it produces the best sweet tea. In addition to using tea bags, the recipe calls for 1/2 cup of sugar for every 1 1/2 cups of water. Write an equation that shows the relationship between the number of cups of sugar to the number of cups of water in this recipe. Let x represent the number of cups of sugar and y represent the number of cups of water
Answer:
y = 3x
Step-by-step explanation:
In this case, the ratio between sugar and water is 0.5 : 1.5 or 1 : 3.
That means that you need three cups of water for every cup of sugar.
In order to put this into an equation, the slope would be 3/1 or simply 3, and the y-intercept is 0. Even if you used the 0.5 : 1.5 ratio, the slope would be the same since 1.5 / 0.5 = 3.
If you place 950 kg of material into a mixer and discharge 895 kg of product, what percentage of the raw materials was discharged
Answer:
94.2%
Step-by-step explanation:
Most of the materials is converted into product.
Assuming the amount of material discharged = the amount of product discharged.
The percentage of material discharged is
(895/950)×100 = 94.2%
If P is the circumcenter △ A B C, and AD =3x - 11, DB =5x - 29, PC =18, find DP.
Answer:
8.2
Step-by-step explanation:
1. Since AD is equal to DB, we will need to set 3x-11 equal to 5x-29. When you solve the equation you will get 9 for x.
2. Plug the 9 in for x in DB (you could plug it in for AD instead if you want since DB and AD are equal to each other). You should get 16.
3. Take a look at the smaller triangle DBP which is part of triangle ABC. We will need to use the Pythagorean Theorem. Square the legs of the triangle and set it equal to the hypotenuse squared. One of the legs, DP, is unknown so we will just have it as x. The other leg is DB, which is 16.
* PB is the hypotenuse and has a value of 18. Remember that P is the circumcenter, which means it is equidistant to each vertex and each angle bisector is equal to each other. As a result PC is equal to PB which happens to be 18.
You should have it something like this:
16^2 + x^2 = 18^2
Now do all the squaring and stuff and you should eventually get x^2 = 68
4. Square root x^2 = 68 and you will get a long decimal. You will need to round it to the tenths place.
5. After rounding you will finally get 8.2
The measurement of DP is 8.2 units.
What is circumcenter of a triangle?The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.
Given that, P is the circumcenter △ ABC, and AD = 3x - 11, DB =5x - 29,
PC = 18
Since, DP bisects AB, so, AB = DP
3x-11 = 5x-29
x = 9
DB = 5*9 - 29 = 16
Since, P is the circumcenter, then PB = PC = PA = 18
Using Pythagoras theorem in Δ BPD
PB² = DB² + DP²
DP² = PB² - DB²
DP² = 18² - 16²
DP² = 324 - 256
DP² = 68
DP = 8.2
Hence, The measurement of DP is 8.2 units.
For more references on circumcenter of a triangle, click;
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4. [-12 Points)
DETAILS
HARMATHAP12 12.4.006.
Cost, revenue, and profit are in dollars and x is the number of units.
If the marginal cost for producing a product is MC = 84 - 4e -0.01x, with a fixed cost of $6,200, find the total cost function.
C(x) =
Find the cost of producing 40 units to the nearest dollar)
$
Need Help?
Read it
Answer:
800
Step-by-step explanation:
i need help please on this math
Answer:
a
Step-by-step explanation:
Answer:
The answer is A
Step-by-step explanation:
If you put in a calculator 4/5 and then 64/80 you get the same decimal meaning that they have the same ratio.
Savannah got her hair done. If it costs $120 and she tips her hairdresser 18%, how much tip will her hairdresser receive
Answer:
21.6
Step-by-step explanation:
Do 0.18 (18%) times 120
Answer:
21.60
Step-by-step explanation:
18% of 120 is 21.60, so we would tip her hairdresser that much :)
Which statement best explains whether the table represents a linear or nonlinear function?
Input Output
(x) (y)
−2 7
−5 4
−8 1
−11 −2
A It is a linear function because there is a constant rate of change in both the input and output values.
B It is a nonlinear function because there is a constant rate of change in both the input and output values.
C It is a linear function because the input values are decreasing.
D It is a nonlinear function because the input values are increasing.
Answer:
b
Step-by-step explanation:
Point Q' is the image of Q(-7,-6) under the translation (x, y) + (x+12, y + 8).
What are the coordinates of Q'?
Answer: ( 5, 2 )
Step-by-step explanation:
If the original image is Q ( -7, -6 ) and you are looking for Q' ( A translated version of the original image )
If the translation is ( x, y ) --> ( x + 12, y + 8 )
If coordinates are ( x , y ) and the original image is ( - 7, -6 )
Then that means:
x = -7
y = -6
Use the original coordinates you have and add them with the translation given to you
( - 7 + 12 ) and ( -6 + 8 )
The coordinates of Q' is ( 5, 2 )
Hope this helps!!
Find the value of log100,000
Answer:
Step-by-step explanation:
5
What is the image of the point (-8,7)(−8,7) after a rotation of 90^ counterclockwise about the origin?
Answer:
(−7,−8)
Step-by-step explanation:
After the data was recorded in the dot plot, another response was found for a participant who watches 30 hours of television each week. How will this additional data point affect the mean number of hours of television watched each week? OA The effect on the mean number of hours cannot be determined. OB. The mean number of hours will remain the same. O c. The mean number of hours will decrease. OD. The mean number of hours will increase. please help
Answer:
I think its A
Step-by-step explanation:
...
Answer:
the answer is D I hope it helps you