Answer with explanation:
It is given that z is an irrational complex number.
Z= x + i y
Where x is real part and y is Imaginary part.x and y can be any Real number.
If z is an irrational complex number , then both real part and imaginary part should be a complex number.That is x, y ∈Q, then ,Q= Set of Irrationals.
It means , x and y both should be an irrational number.
So, if x is any irrational number then, [tex]\sqrt{x}[/tex] will be also an irrational number.
There are ten members of the debate team: Ann, Bill, Carla, Don, Ed, Frank, Gretchen, Harry, Isabel, and Jen. If they randomly choose three co-leaders of the group, what is the probability that Gretchen, Don, and Carla are chosen as the co-leaders?
Answer:
1/120
Step-by-step explanation:
There are 10 members, and three are chose as co-leaders. The number of possible combinations is:
₁₀C₃ = 120
One of those 120 combinations is Gretchen, Don, and Carla. So the probability is 1/120, or approximately 0.83%.
Factoring out the GCF
8m^2n^3-24m^2n^2 +4m^3 n
Answer:
[tex]\large\boxed{8m^2n^3-24m^2n^2+4m^3n=4m^2n(2n^2-6n+m)}[/tex]
Step-by-step explanation:
[tex]8m^2n^3-24m^2n^2+4m^3n\\\\8m^2n^3=\boxed{(2)}\boxed{(2)}(2)\boxed{(m)}\boxed{(m)}\boxed{(n)}(n)(n)\\\\24m^2n^2=\boxed{(2)}\boxed{(2)}(2)(3)\boxed{(m)}\boxed{(m)}\boxed{(n)}(n)\\\\4m^3n=\boxed{(2)}\boxed{(2)}\boxed{(m)}\boxed{(m)}(m)\boxed{(n)}\\\\8m^2n^3-24m^2n^2+4m^3n\\\\=\boxed{(2)}\boxed{(2)}\boxed{(m)}\boxed{(m)}\boxed{(n)}\bigg((2)(n)(n)-(2)(3)(n)+(m)\bigg)\\\\=4m^2n(2n^2-6n+m)[/tex]
1. If $18,000 is invested at 6% compounded monthly, what is the amount after 7 years?
Quadrilateral ABCD has vertices A(-3, 4), B(1, 3), C(3, 6), and D(1, 6). Match each set of vertices of quadrilateral EFGH with the transformation that shows it is congruent to ABCD. E(-3, -4), F(1, -3), G(3, -6), and H(1, -6) a translation 7 units right E(-3, -1), F(1, -2), G(3, 1), and H(1, 1) a reflection across the y-axis E(3, 4), F(-1, 3), G(-3, 6), and H(-1, 6) a reflection across the x-axis E(4, 4), F(8, 3), G(10, 6), and H(8, 6)
Answer:
The set of vertices of quadrilateral EFGH with the transformation 7 units right is E(4 , 4) , F(8 , 3) , G(10 , 6) , and H(8 , 6)
The set of vertices of quadrilateral EFGH with a reflection across the y-axis is E(3 , 4) , F(-1 , 3) , G(-3 , 6) , and H(-1 , 6)
The set of vertices of quadrilateral EFGH with a reflection across the x-axis is E(-3 , -4) , F(1 , -3) , G(3 , -6) , and H(1 , -6)
Step-by-step explanation:
Lets revise some transformation
- If point (x , y) reflected across the x-axis
then Its image is (x , -y)
- If point (x , y) reflected across the y-axis
then Its image is (-x , y)
- If the point (x , y) translated horizontally to the right by h units
then its image is (x + h , y)
- If the point (x , y) translated horizontally to the left by h units
then its image is (x - h , y)
* Now lets solve the problem
- The vertices of the quadrilateral ABCD are:
A = (-3 , 4) , B = (1 , 3) , C = (3 , 6) , D = (1 , 6)
- The quadrilateral ABCD translated 7 units right to form
quadrilateral EFGH
- We add each x-coordinates in ABCD by 7
∵ A = (-3 , 4)
∴ E = (-3 + 7 , 4) = (4 , 4)
∵ B = (1 , 3)
∴ F = (1 + 7 , 3) = (8 , 3)
∵ C = (3 , 6)
∴ G = (3 + 7 , 6) = (10 , 6)
∵ D = (1 , 6)
∴ H = (1 + 7 , 6) = (8 , 6)
* The set of vertices of quadrilateral EFGH with the transformation
7 units right is E(4 , 4) , F(8 , 3) , G(10 , 6) , and H(8 , 6)
- The quadrilateral ABCD reflected across the y-axis to form
quadrilateral EFGH
- We change the sign of the x-coordinate
∵ A = (-3 , 4)
∴ E = (3 , 4)
∵ B = (1 , 3)
∴ F = (-1 , 3)
∵ C = (3 , 6)
∴ G = (-3 , 6)
∵ D = (1 , 6)
∴ H = (-1 , 6)
* The set of vertices of quadrilateral EFGH with a reflection across the
y-axis is E(3 , 4) , F(-1 , 3) , G(-3 , 6) , and H(-1 , 6)
- The quadrilateral ABCD reflected across the x-axis to form
quadrilateral EFGH
- We change the sign of the y-coordinate
∵ A = (-3 , 4)
∴ E = (-3 , -4)
∵ B = (1 , 3)
∴ F = (1 , -3)
∵ C = (3 , 6)
∴ G = (3 , -6)
∵ D = (1 , 6)
∴ H = (1 , -6)
* The set of vertices of quadrilateral EFGH with a reflection across the
x-axis is E(-3 , -4) , F(1 , -3) , G(3 , -6) , and H(1 , -6)
Answer:
here is the answer
Step-by-step explanation:
If records indicate that 15 houses out of 1000 are expected to be damaged by fire in any year, what is the probability that a woman who owns 14 houses will have fire damage in 2 of them in a year? (Round your answer to five decimal places.)
Answer: 0.01708
Step-by-step explanation:
Given : If records indicate that 15 houses out of 1000 are expected to be damaged by fire in any year.
i.e. the probability that house damaged buy fire in a year : [tex]p=\dfrac{15}{1000}=0.015[/tex]
The formula for binomial distribution is given by :-
[tex]^{n}C_xp^x(1-p)^{n-x}[/tex]
Now, the probability that a woman who owns 14 houses will have fire damage in 2 of them in a year (put n=14 and x=2), we get
[tex]^{14}C_2(0.015)^2(1-0.015)^{14-2}\\\\=\dfrac{14!}{2!(14-2)!}(0.015)^2(0.985)^{12}\\\\=0.0170788520518\approx0.01708[/tex]
Hence, the required probability = 0.01708
Eliminate all exponents by Expanding 6^3 y^4
Answer:
216*y*y*y*y
Step-by-step explanation:
6 cubed is 216, and y^4 expanded is yyyy. So if I'm understanding correctly, you want as your answer:
216*y*y*y*y
In 1898, L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.61. (a) What is the probability of more than 1 death in a corps in a year
Answer: Hence, Probability of more than 1 death in a corps in a year is 0.126.
Step-by-step explanation:
Since we have given that
Mean for a poisson distribution (λ) = 0.61
Number of years = 20 years
We need to find the probability of more than 1 death in a corps in a year.
P(X>1)=1-P(X=0)-P(X=1)
Here,
[tex]P(X=0)=\dfrac{e^{-0.61}(0.61)^0}{0!}=0.543\\\\and\\\\P(X=1)=\dfrac{e^{-0.61}(0.61)}{1}=0.331[/tex]
So,
P(X>1)=1-0.543-0.331=0.126
Hence, Probability of more than 1 death in a corps in a year is 0.126.
Using the Poisson distribution with a mean of 0.61, we calculate the probability of 0 or 1 death and subtract that from 1 to get the probability of more than 1 death in a Prussian cavalry corps in a year.
Explanation:Calculating Probability Using the Poisson DistributionBased on L. J. Bortkiewicz's study, the number of soldiers killed by horse kicks in the Prussian cavalry follows a Poisson distribution with a mean (λ) of 0.61. To calculate the probability of more than one death in a corps in a year, we use the Poisson probability formula:
P(X > k) = 1 - P(X ≤ k)
Where P(X > k) is the probability of having more than k events (in this case, deaths), and P(X ≤ k) is the probability of k or fewer events. In this scenario, k equals 1. So, we need to calculate the probability of 0 or 1 death and subtract from 1 to get the probability of more than 1 death.
Using the Poisson probability formula:
Calculate P(X = 0) and P(X = 1).Add P(X = 0) and P(X = 1) to get P(X ≤ 1).Subtract P(X ≤ 1) from 1 to obtain P(X > 1).Let's calculate:
P(X = 0) = (e^(-0.61))*(0.61^0)/(0!) = e^(-0.61)P(X = 1) = (e^(-0.61))*(0.61^1)/(1!) = 0.61*e^(-0.61)P(X ≤ 1) = P(X = 0) + P(X = 1)P(X > 1) = 1 - P(X ≤ 1)The resulting calculation will give us the probability of more than one death due to horse kicks in a Prussian cavalry corps within one year.
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An automobile tire has a radius of 0.315 m, and its center moves forward with a linear speed of v = 19.3 m/s. (a) Determine the angular speed of the wheel. (b) Relative to the axle, what is the tangential speed of a point located 0.193 m from the axle?
Answer:
angular speed: 61.3 radians/stangential speed at .193 m: 11.8 m/sStep-by-step explanation:
The forward speed of the center of the tire with respect to the ground is the same as the tangential speed of the tire at its full radius of 0.315 m, relative to the axle.
The angular speed of the tire is the ratio of tangential speed to radius:
(19.3 m/s)/(0.315 m) ≈ 61.27 radians/s
The tangential speed at any other radius is the product of angular speed and radius. At a radius of 0.193 m, the tangential speed is ...
(0.193 m)×(19.3 m/s)/(0.315 m) ≈ 11.825 m/s ≈ 11.8 m/s
(a) The angular speed of the wheel is approximately 61.27 rad/s.
(b) The tangential speed of a point 0.193 m from the axle is about 11.82 m/s, relative to the axle.
let's break this down step by step.
Given:
Radius of the tire, r = 0.315 m
Linear speed of the center of the tire, v = 19.3 m/s
Distance from the axle to the point, d = 0.193 m
(a) To determine the angular speed of the wheel (ω), we can use the formula relating linear speed (v) and angular speed (ω) for a rotating object:
v = ω * r
where:
v = linear speed
ω = angular speed
r = radius
We can rearrange this equation to solve for ω:
ω = v / r
Now, substitute the given values:
ω = 19.3 m/s / 0.315 m
ω ≈ 61.27 rad/s
So, the angular speed of the wheel is approximately 61.27 rad/s.
(b) To find the tangential speed of a point located 0.193 m from the axle relative to the axle, we'll use the formula:
Tangential speed (vt) = ω x distance from the axle (d)
We already have the value of ω from part (a), which is approximately 61.27 rad/s. Now, let's calculate the tangential speed:
vt = 61.27 rad/s x 0.193 m
vt ≈ 11.82 m/s
So, the tangential speed of a point located 0.193 m from the axle, relative to the axle, is approximately 11.82 m/s.
How do you solve this system by substitution?
Answer is 'a'.(4;8;8)
All the details are provided in the attachment; the answer is marked with green colour.
The sum of Eli’s age and Cecil’s age is 60. Six years ago, Eli was three times old as Cecil. Find Eli’s age now
Answer:
Eli's age = 42 years
Step-by-step explanation:
Let x be Eli's age and y be Cecil's age
So,
According to the statement given
x+y=60 eqn 1
Eli's age 6 years ago = x-6
Cecil's age 6 years ago = y-6
So according to the given statement
x-6 = 3(y-6)
x-6 = 3y - 18
x-3y = -18+6
x-3y= -12 eqn 2
Subtracting eqn 2 from eqn 1
x+y - (x-3y) = 60 - (-12)
x+y-x+3y = 60+12
4y = 72
y = 18
Cecil's age = 18 years
Putting y = 18 in eqn 1
x+18=60
x = 60-18
x = 42
Eli's age = 42 years ..
×
_+4=20
6
Solve the following equation. Then place the correct number in the box provided.
Answer:
Step-by-step explanation:
X/6 +4 = 20
X/6 = 20 - 4
X/6 = 16
X = 16/6
<c is opposite
please help me find the answer thank you :)
Answer:
A) [tex]\angle C \text{ is opposite } \overline{AB}[/tex]
Step-by-step explanation:
We need to name the side of the triangle that is across from angle C. A side of a triangle, if not otherwise given, is named based on the two points that form it. In this case, the side is formed by points B and C, so the side is called [tex]\overline{BC}[/tex].
A sample of size n=10n=10 is drawn from a population. The data is shown below.
138.8
116.4
97
114.2
138.8
100.4
128.8
138.8
101.9
117.6
What is the range of this data set?
range =
What is the standard deviation of this data set? (Remember, it is a sample.) Please report the answer with appropriate rounding, rounding to two more decimal places than the original data. Please, please, please do not calculate the value by hand.
stdev =
Answer:
The range of this data set is 41.8
The standard deviation of the data set is 16.42
Step-by-step explanation:
* Lets read the information and use it to solve the problem
- There is a sample of size n = 10, is drawn from a population
- The data are: 97 , 100.4 , 101.9 , 114.2 , 116.4 , 117.6 , 128.8 , 138.8 ,
138.8 , 138.8
- The range is the difference between the largest number and
the smallest number
∵ The largest number is 138.8
∵ The smallest number is 97
∴ The range = 138.8 - 97 = 41.8
* The range of this data set is 41.8
- Lets explain how to find the standard deviation
# Step 1: find the mean of the data set
∵ The mean = the sum of the data ÷ the number of the data
∵ The data set is 97 , 100.4 , 101.9 , 114.2 , 116.4 , 117.6 , 128.8 , 138.8 ,
138.8 , 138.8
∵ Their sum = 97 + 100.4 + 101.9 + 114.2 + 116.4 + 117.6 + 128.8 + 138.8 +
138.8 + 138.8 = 1192.7
∵ n = 10
∴ The mean = 1192.7 ÷ 10 = 119.27
# Step 2: subtract the mean from each data and square the answer
∴ (97 - 119.27)² = 495.95
∴ (100.4 - 119.27)² = 356.08
∴ (101.9 - 119.27)² = 301.72
∴ (114.2 - 119.27)² = 25.70
∴ (116.4 - 119.27)² = 8.24
∴ (117.6 - 119.27)² = 2.79
∴ (128.8 - 119.27)² = 90.82
∴ (138.8 - 119.27)² = 381.42
∴ (138.8 - 119.27)² = 381.42
∴ (138.8 - 119.27)² = 381.42
# Step 3: find the mean of these squared difference
∵ A Sample: divide by n - 1 when calculating standard deviation of
a sample
∵ The mean = the sum of the data ÷ (the number of the data - 1)
∵ The sum = 495.95 + 356.08 + 301.72 + 25.70 + 8.24 + 2.79 + 90.82 +
381.42 + 381.42 + 381.42 = 2425.56
∴ The mean = 2425.56 ÷ (10 - 1) = 269.51
# Step 4: the standard deviation is the square root of this mean
∴ The standard deviation = √(269.51) = 16.416658 ≅ 16.42
* The standard deviation of the data set is 16.42
Consider the function f(x)=2x^3+24x^2−54x+9,−9≤x≤2 This function has an absolute minimum value equal to? and an absolute maximum value equal to ?
If you evaluate it, it's −9≤x≤2 and trying to find the absolute maximum/minimum of it then you'll get nothing due to it being an improper fraction of some sorts.. And there's still nothing when trying to find it all together.. Sorry that I wasn't that much help.
Please show me steps to find 1/3 + m = 13/15
Hello!
Answer:
[tex]\boxed{m=\frac{8}{15}}[/tex]
Step-by-step explanation:
First, you switch sides.
[tex]m+\frac{1}{3}=\frac{13}{15}[/tex]
Then, you subtract by 1/3 from both sides.
[tex]m+\frac{1}{3}-\frac{1}{3}=\frac{13}{15}-\frac{1}{3}[/tex]
Simplify and solve.
[tex]\frac{13}{15}=\frac{8}{15}[/tex]
Therefore, [tex]\boxed{\frac{8}{15}}[/tex], which is our final answer.
I hope this helps you!
Have a nice day! :)
Help on this ALGEBRA QUESTIONS !!!
Simplify the expression, if possible. 512 ^1/2
A. 32
B. 16√ 2
C. 64
D. It's not a real number.
Note that [tex]x^{\frac{1}{2}}=\sqrt[2]{x}[/tex]
Which means that:
[tex]512^{\frac{1}{2}}=\sqrt[2]{512}=\sqrt[2]{16^2\cdot2}=\boxed{16\sqrt[2]{2}}[/tex]
the answer is B.
Hope this helps.
r3t40
a.
solve
[tex]\frac{1}{n} \pi = \theta - \frac{1}{2}sin(2 \theta)[/tex] for [tex] \theta[/tex] in terms of "n"
(derivation of equation below)
b. Based on your answer in
part a, if [tex] \theta = arccos(1 - \frac{a}{r} ) = {cos}^{ - 1} (1 - \frac{a}{r} )[/tex] or [tex] a = r-2cos( \theta)[/tex]
find "a" as a function of
r & n. (find f(r,n)=a).
alternately, if a+b=r, we can write [tex] \theta = arccos( \frac{b}{r} ) = {cos}^{ - 1} (\frac{b}{r} )[/tex]
then solve for "a" in terms of r and n
show all work and reasoning.
Solve analytically if possible
Answer:
a) There is no algebraic method for finding θ in terms of n
b) should be a = r(1 -cos(θ))
Step-by-step explanation:
Algebraic methods have been developed for solving trig functions and polynomial functions individually, but not in combination. In general, the solution is easily found numerically, but not analytically.
You would be looking for the numerical solution to ...
f(n, θ) = 0
where f(n, θ) can be ...
f(n, θ) = θ - (1/2)sin(2θ) - π/n
___
The attached shows Newton's method iterative solutions for n = 3 through 6:
for n = 3, θ ≈ 1.3026628373
for n = 4, θ ≈ 1.15494073001
...
A car company claims that the mean gas mileage for its luxury sedan is at least 24 miles per gallon. A random sample of 7 cars has a mean gas mileage of 23 miles per gallon and a standard deviation of 1.1 miles per gallon. At α=0.05, can you support the company’s claim assuming the population is normally distributed?
Answer:
a) t sampling distribution because B the population is normal, and standard deviation is unknown
b) H0: mu <= 21
HA mu > 21
alpha = 0.05
t critical value at 4 df and alpha 0.05 is 2.132
The rejection region is t > 2.132
t = (xbar - µ)/(s/√n)
t = (19 - 21 )/(4/√5)
t = -2 / (4/2.2361)
t = -1.118
t does not fall into the rejection region, so we have insufficient evidence to reject the null hypothesis. The claim cannot be verified.
We tested the manufacturer's claim that the mean mpg is greater than 21, at alpha = 0.05. We used a one-tailed one-sample t-test (4 df). We placed the rejection region in the right tail of the t-distribution because we were only interested in the claim that the mileage was more than 21. The test result showed that the claim could not be validated. The sample mean was 19, which was less than the claim, so no calculations were needed to reject the null hypothesis. We were not able to find that the mean was statistically greater than 21.
Based on hypothesis testing in statistics, there isn't enough evidence to support the car company’s claim that the average gas mileage for its luxury sedan is at least 24 miles per gallon.
Explanation:This question involves the use of hypothesis testing in statistics. The null hypothesis for this test is that the mean gas mileage is at least 24 miles per gallon (μ >= 24), and the alternative hypothesis is that the mean gas mileage is less than 24 miles per gallon (μ < 24).
With a calculated sample mean of 23 miles per gallon and a sample standard deviation of 1.1 miles per gallon for 7 cars, we use the standard error formula SE = σ/√n = 1.1/√7 = 0.415 to calculate the standard error. The t value is then calculated as (X - μ) / SE = (23 - 24) / 0.415 = -2.41.
Using a t-distribution table, we find that the critical value for a one-tailed test with degrees of freedom = n - 1 = 6 and α = 0.05 is -1.943. Since our calculated t value (-2.41) is less than the critical value (-1.943), we reject the null hypothesis. Therefore, we cannot support the company’s claim that the mean gas mileage for its luxury sedan is at least 24 miles per gallon.
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Use the Newton-Raphson method to find the root of the equation f(x) = In(3x) + 5x2, using an initial guess of x = 0.5 and a stopping criterion of &= 0.1%. Report your answer to 3 decimal places. What is the estimate for the root? What is the approximate relative error? %
Answer with explanation:
The equation which we have to solve by Newton-Raphson Method is,
f(x)=log (3 x) +5 x²
[tex]f'(x)=\frac{1}{3x}+10 x[/tex]
Initial Guess =0.5
Formula to find Iteration by Newton-Raphson method
[tex]x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}\\\\x_{1}=x_{0}-\frac{f(x_{0})}{f'(x_{0})}\\\\ x_{1}=0.5-\frac{\log(1.5)+1.25}{\frac{1}{1.5}+10 \times 0.5}\\\\x_{1}=0.5- \frac{0.1760+1.25}{0.67+5}\\\\x_{1}=0.5-\frac{1.426}{5.67}\\\\x_{1}=0.5-0.25149\\\\x_{1}=0.248[/tex]
[tex]x_{2}=0.248-\frac{\log(0.744)+0.30752}{\frac{1}{0.744}+10 \times 0.248}\\\\x_{2}=0.248- \frac{-0.128+0.30752}{1.35+2.48}\\\\x_{2}=0.248-\frac{0.17952}{3.83}\\\\x_{2}=0.248-0.0468\\\\x_{2}=0.2012[/tex]
[tex]x_{3}=0.2012-\frac{\log(0.6036)+0.2024072}{\frac{1}{0.6036}+10 \times 0.2012}\\\\x_{3}=0.2012- \frac{-0.2192+0.2025}{1.6567+2.012}\\\\x_{3}=0.2012-\frac{-0.0167}{3.6687}\\\\x_{3}=0.2012+0.0045\\\\x_{3}=0.2057[/tex]
[tex]x_{4}=0.2057-\frac{\log(0.6171)+0.21156}{\frac{1}{0.6171}+10 \times 0.2057}\\\\x_{4}=0.2057- \frac{-0.2096+0.21156}{1.6204+2.057}\\\\x_{4}=0.2057-\frac{0.0019}{3.6774}\\\\x_{4}=0.2057-0.0005\\\\x_{4}=0.2052[/tex]
So, root of the equation =0.205 (Approx)
Approximate relative error
[tex]=\frac{\text{Actual value}}{\text{Given Value}}\\\\=\frac{0.205}{0.5}\\\\=0.41[/tex]
Approximate relative error in terms of Percentage
=0.41 × 100
= 41 %
2x + 1 < 5
Solve the following inequality. Then place the correct number in the box provided.
Answer:
[tex]\boxed{x<2}[/tex]
Step-by-step explanation:
You subtract by 1 from both sides of equation.
[tex]2x+1-1<5-1[/tex]
Simplify.
[tex]5-1=4[/tex]
[tex]2x<4[/tex]
Divide by 2 from both sides of equation.
[tex]\frac{2x}{2}<\frac{4}{2}[/tex]
Simplify, to find the answer.
[tex]4\div2=2[/tex]
X<2 is the correct answer.
Answer: [tex]x<2[/tex]
Step-by-step explanation:
Given the inequality [tex]2x + 1 < 5[/tex] you can follow this procedure to solve it:
The first step is to subtract 1 from both sides on the inequaltity.
[tex]2x + 1-(1) < 5-(1)\\\\2x < 4[/tex]
Now, the second and final step is to divide both sides of the inequality by 2. Therefore, you get this result:
[tex]\frac{2x}{2}<\frac{4}{2} \\\\(1)x<2\\\\x<2[/tex]
In how many ways can 4 married couples attending a concert be seated in a row of 8 seats if: a) There are no restrictions? b) Each married couple is seated together?
Answer:
a) 40,320
b) 384
Step-by-step explanation:
Given,
The total number of seats = 8,
Also, these 8 seats are occupied by 4 married couples or 8 people,
a) Thus, if there is no restrictions of seating ( that is any person can seat with any person ),
Then, the total number of arrangement = 8 ! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40320,
b) if each married couple is seated together,
Then, the 4 couples can seat in 4 pair of seats,
Also, in a pair of seats a couple can choose any of the two seats,
So, the total number of arrangement
[tex]=4! \times 2^4[/tex]
[tex]=24\times 16[/tex]
[tex]=384[/tex]
Write an exponential function y = abx for a graph that includes (–4, 72) and (–2, 18).
Answer:
[tex]y=4.5(0.5)^{x}[/tex]
Step-by-step explanation:
* Lets revise the meaning of exponential function
- The form of the exponential function is [tex]y=ab^{x}[/tex],
where a ≠ 0, b > 0 , b ≠ 1, and x is any real number
- It has a constant base b
- It has a variable exponent x
- To solve an exponential equation, take the log or ln of both sides,
and solve for the variable
* Lets solve the problem
∵ y = a(b)^x is an exponential function
∵ Its graph contains the point (-4 , 72) and (-2 , 18)
- Lets substitute x and y by the coordinates of these points
# Point (-4 , 72)
∵ [tex]y=ab^{x}[/tex]
∵ x = -4 and y = 72
∴ [tex]72=ab^{-4}[/tex]
- The change any power from -ve to +ve reciprocal the base of
the power ([tex]p^{-n}=\frac{1}{p^{n}}[/tex]
∴ [tex]72=\frac{a}{b^{4}}[/tex]
- By using cross multiplication
∴ [tex]a=72b^{4}[/tex] ⇒ (1)
# Point (-2 , 18)
∵ x = -2 and y = 18
∴ [tex]18=ab^{-2}[/tex]
∴ [tex]18=\frac{a}{b^{2}}[/tex]
- By using cross multiplication
∴ a = 18b² ⇒ (2)
- Equate the two equations (1) and (2)
∴ [tex]72b^{4}=18b^{2}[/tex]
- Divide both sides by 18b²
∵ [tex]\frac{72b^{4}}{18b^{2}}=4b^{4-2}=4b^{2}[/tex]
∵ [tex]\frac{18b^{2}}{18b^{2}}=(1)b^{2-2}=(1)b^{0}=(1)(1)=1[/tex]
∴ 4b² = 1 ⇒ divide both sides by 4
∴ [tex]b^{2}=\frac{1}{4}=0.25[/tex] ⇒ take square root for both sides
∴ b = √0.25 = 0.5
- Lets substitute the value ob b in equation (1) or (2) to find a
∵ a = 18b²
∵ b² = 0.25
∴ a = 18(0.25) = 4.5
- Lets substitute the values of a and b in the equation [tex]y=ab^{x}[/tex]
∴ [tex]y=4.5(0.5)^{x}[/tex]
- We can write it using fraction
∴ [tex]y=\frac{9}{2}(\frac{1}{2})^{x}[/tex]
ANSWER
[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x}[/tex]
EXPLANATION
Let the exponential function be
[tex]y = a {b}^{x} [/tex]
Since the graph includes (–4, 72), it must satisfy this equation.
[tex]72= a { b}^{ - 4}[/tex]
Multiply both sides by b⁴ .
This implies that,
[tex]a = 72 {b}^{4} ...1[/tex]
The graph also includes (-2,18).
We substitute this point also to get:
[tex]18=a {b}^{ - 2} [/tex]
Multiply both sides by b²
[tex]a = 18 {b}^{2} ...(2)[/tex]
We equate (1) and (2) to obtain:
[tex]72 {b}^{4} = 18 {b}^{2} [/tex]
Multiply both sides by
[tex] \frac{72 {b}^{4} }{ {18b}^{4} } = \frac{18 {b}^{2} }{18 {b}^{4} } [/tex]
[tex]4 = \frac{1}{ {b}^{2} } [/tex]
Or
[tex]{2}^{ 2} = ( \frac{1}{b} )^{2} [/tex]
[tex] \frac{1}{b} = 2[/tex]
[tex]b = \frac{1}{2} [/tex]
Put b=½ into equation (2).
[tex]a = 18 {( \frac{1}{2} })^{2} [/tex]
[tex]a = \frac{18}{4} [/tex]
[tex]a = \frac{9}{2} [/tex]
Therefore the equation is
[tex]y = \frac{9}{2} ( { \frac{1}{2} })^{x} [/tex]
8.) Out of 600 employees, only 270 are happy with their pay. What percent is this? (Please show your work.) *
For this case we can raise a rule of three:
600 employees -------------> 100%
270 employees -------------> x
Where the variable "x" represents the percentage of employees who are satisfied with their salary. So, we have:
[tex]x = \frac {270 * 100} {600}\\x = 45[/tex]
Thus, 45% of employees are satisfied with their salary.
Answer:
45%
ANSWER
[tex]45\%[/tex]
EXPLANATION
The total number of employees is 600.
The number of employees who are happy with their pay is 270.
The percentage of employees who are happy with their pay is the number who are happy with their pay divided by total number of employees times 100%
[tex] \frac{270}{600} \times 100\%[/tex]
This simplifies to
[tex]45\%[/tex]
According to an article in Newsweek, the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100: 114 (46.7% girls). Suppose you don’t believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7?
Based on the study results, the percentage of girls born in China could range from 32.98% to 47.02%.
Explanation:To determine if the percentage of girls born in China is 46.7%, we can calculate the confidence interval for the proportion of girls in the population using a binomial distribution. Based on the study, out of 150 births, 60 were girls and 90 were boys.
Calculate the sample proportion of girls: p = 60/150 = 0.4Calculate the standard error: SE = sqrt((p*(1-p))/n) = sqrt((0.4*(1-0.4))/150) ≈ 0.0357Calculate the margin of error: ME = z*(SE) = 1.96*(0.0357) ≈ 0.0702Calculate the confidence interval: CI = p ± ME = 0.4 ± 0.0702 = (0.3298, 0.4702)The confidence interval suggests that the true proportion of girls born in China could range from 32.98% to 47.02%. Since the reported figure of 46.7% falls within this interval, it is plausible based on the study results.
Choose the equation of the graph shown.
y=-2cosx
y=-2sinx
y=2cosx
y= 2sinx
Answer:
y = -2 sin x
Step-by-step explanation:
As a basic,
y = cos x has a value of 1 at x = 0, and
y = sin x has a value of 0 at x = 0
Note: the value 1 can change to 2, 3, 4, etc. if the amplitude increases
Looking at the graph at x = 0, we see the y-value is 0, so definitely this is a sin graph. We can eliminate the cos choices.
So is it y = 2 sin x or y = -2 sin x??
If the graph goes downward from 0 (at x = 0), it is reflected of original, so that would be y = - sinx.
Since the graph decreases (goes downward) from x = 0, it is definitely the graph of negative sin. So y = - 2 sin x
Captain Ralph is in trouble near the sunny side of Mercury. The temperature of the ship's hull when he is at location (x, y, z) will be given by T (x, y, z) = e−x2 − 2y2 − 3z2, where x, y, and z are measured in meters. He is currently at (1, 1, 1). (a) In what direction should he proceed in order to decrease the temperature most rapidly?
The maximum rate of change occurs in the direction of the gradient vector at (1, 1, 1).
[tex]T(x,y,z)=e^{-x^2-2y^2-3z^2}\implies\nabla T(x,y,z)=\langle-2x,-4y,-6z\rangle e^{-x^2-2y^2-3z^2}[/tex]
At (1, 1, 1), this has a value of
[tex]\nabla T(1,1,1)=\langle-2,-4,-6\rangle e^{-6}[/tex]
so the captain should move in the direction of the vector [tex]\langle-1, -2, -3\rangle[/tex] (which is a vector pointing in the same direction but scaled down by a factor of [tex]2e^{-6}[/tex]).
The direction Captain Ralph should proceed in order to decrease the temperature most rapidly is towards the direction of the steepest temperature decrease gradient. This direction is given by the negative gradient of the temperature function.
In this case, the negative gradient of T(x, y, z) = e^(-x^2 - 2y^2 - 3z^2) at the point (1, 1, 1) would be (-2e^(-6), -4e^(-6), -6e^(-6)).
Therefore, Captain Ralph should proceed in the direction (-2e^(-6), -4e^(-6), -6e^(-6)) to decrease the temperature most rapidly at his current location.
(05.05 MC) The area of a triangle is 24 square inches. What is the height of the triangle if the base length is 8 inches?
height: 6
formula: 1/2bh
hope this helps :)
For this case we have that by definition, the area of a triangle is given by:
[tex]A = \frac {1} {2} b * h[/tex]
Where:
b: It's the base
h: It's the height
They tell us as data that:
[tex]A = 24 \ in ^ 2\\b = 8in[/tex]
Substituting the data and clearing the height:
[tex]24 = \frac {1} {2} 8 * h\\24 = 4h\\h = \frac {24} {4}\\h = 6[/tex]
So, the height of the triangle is 6in
Answer:
[tex]h = 6in[/tex]
Plz help out!! honest answer!
Answer:
2nd choice: Counterclockwise rotation about the origin by 180 degrees followed by a reflection about the y-axis
Step-by-step explanation:
A simple reflection across the x-axis will do.
A rotation of 180 degrees about the origin is equivalent to a reflection across both axes. Then a reflection back across the y-axis leaves the net effect being the desired reflection across the x-axis.
Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y, z) = xe^y + ye^z + ze^x, (0, 0, 0), v = 6, 3, −3
Final answer:
The directional derivative of the function f(x, y, z) = [tex]xe^y + ye^z + ze^x[/tex] at the point (0, 0, 0) in the direction of the vector v = 6, 3, −3 is 0.
Explanation:
To find the directional derivative of the function f(x, y, z) = [tex]xe^y + ye^z + ze^x[/tex] at the point (0, 0, 0) in the direction of the vector v = 6, 3, −3, we first need to find the gradient of f. The gradient of f, denoted as ∇f, is a vector of partial derivatives with respect to each variable. We calculate the partial derivatives as follows:
∂f/∂x = [tex]e^y + ze^x[/tex]∂f/∂y = [tex]xe^y + e^z[/tex]∂f/∂z = [tex]ye^z + xe^x[/tex]At the point (0, 0, 0), the gradient ∇f is (0 + 0, 0 + 1, 0 + 1) = (0, 1, 1).
Next, we need to normalize the given vector v. The normalization process involves dividing v by its magnitude to obtain a unit vector u in the direction of v. The magnitude of v is √(6² + 3² + (-3)²) = √(36 + 9 + 9) = √54. Therefore, the unit vector u is (6/√54, 3/√54, -3/√54).
Finally, the directional derivative of f at (0, 0, 0) in the direction of v is the dot product of ∇f and u, which is (0, 1, 1) ⋅ (6/√54, 3/√54, -3/√54) = 0*6/√54 + 1*3/√54 + 1*(-3)/√54 = 0.
A skateboarder went a kilometer in 3 minutes with the wind, and returned in four minutes against the wind.
How fast could he skateboard one kilometer if there was no wind?
Please explain your answer! Please!
Step-by-step answer:
Answer to problems of this kind is the reciprocal of the harmonic mean of the time required.
We need to find the average of the speeds, not the average of the time.
The respective speeds are 1/3 and 1/4.
The average of the speeds is therefore (1/3+1/4)/2 = 7/24 (harmonic mean of the time taken).
The time required is therefore the reciprocal of the unit speed,
T = 1/(7/24) = 24/7 = 3 3/7 minutes, or approximately 3.43 minutes.