Answer:
cool
Step-by-step explanation:
Find the area under the curve y =f( x) on [a,b] given f(x)=tan(3x) where a=0 b=pi/12
Answer:
The area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.
Step-by-step explanation:
The given function is
[tex]f(x)=\tan(3x)[/tex]
where a=0 and b=pi/12.
The area under the curve y=f(x) on [a,b] is defined as
[tex]Area=\int_{a}^{b}f(x)dx[/tex]
[tex]Area=\int_{0}^{\frac{\pi}{12}}\tan (3x)dx[/tex]
[tex]Area=\int_{0}^{\frac{\pi}{12}}\frac{\sin (3x)}{\cos (3x)}dx[/tex]
Substitute cos (3x)=t, so
[tex]-3\sin (3x)dx=dt[/tex]
[tex]\sin (3x)dx=-\frac{1}{3}dt[/tex]
[tex]a=\cos (3(0))=1[/tex]
[tex]b=\cos (3(\frac{\pi}{12}))=\frac{1}{\sqrt{2}}[/tex]
[tex]Area=-\frac{1}{3}\int_{1}^{\frac{1}{\sqrt{2}}}\frac{1}{t}dt[/tex]
[tex]Area=-\frac{1}{3}[\ln t]_{1}^{\frac{1}{\sqrt{2}}[/tex]
[tex]Area=-\frac{1}{3}(\ln \frac{1}{\sqrt{2}}-\ln (1))[/tex]
[tex]Area=-\frac{1}{3}(\ln 1-\ln \sqrt{2}-0)[/tex]
[tex]Area=-\frac{1}{3}(-\ln 2^{\frac{1}{2}})[/tex]
[tex]Area=-\frac{1}{3}(-\frac{1}{2}\ln 2)[/tex]
[tex]Area=-\frac{1}{6}\ln 2[/tex]
Therefore the area under the curve y=f(x) on [a,b] is [tex]\frac{1}{6}\ln(2)[/tex] square units.
A gas storage tank is in the shape of a right circular cylinder that has a radius of the base of 2ft and a height of 3ft. The farmer wants to paint the tank including both bases but only has 1 gallon of paint. If 1 gallon of paint will cover 162 square feet, will the farmer have enough paint to complete the job?
Answer:
Yes, the farmer have enough paint to complete the job.
Step-by-step explanation:
It is given that a gas storage tank is in the shape of a right circular cylinder.
The radius of the base is 2 ft and the height of cylinder is 3 ft.
The total surface area of a cylinder is
[tex]S=2\pi rh+2\pi r^2[/tex]
Total surface area of gas storage tank is
[tex]S=2\pi (2)(3)+2\pi (2)^2[/tex]
[tex]S=12\pi+8\pi[/tex]
[tex]S=20\pi[/tex]
[tex]S=62.8318530718[/tex]
[tex]S\approx 62.83[/tex]
The total surface area of gas storage tank is 62.83 square feet.
The farmer has 1 gallon of paint and 1 gallon of paint will cover 162 square feet.
Since 62.83<162, therefore 1 gallon of paint is enough to paint the gas storage.
Hence the required statement is Yes, the farmer have enough paint to complete the job.
Answer:
Yes, the farmer have enough paint to complete the job.
Step-by-step explanation:
1 gallon is good
a ball is shot straight upward. with it's height, in feet, after t seconds given by the function f(t)=-16t^2+192t. Find the average velocity of the ball from t=1 to t=6
ANSWER
[tex]80 {ms}^{ - 1} [/tex]
EXPLANATION
The average velocity of the ball is the rateof displacement over the total time.
The height of the ball, in feet, after t seconds is given by the function:
[tex]f(t)=-16t^2+192t[/tex]
At time t=1, the height of the ball is
[tex]f(1)=-16(1)^2+192(1)[/tex]
[tex]f(1)=-16+192 = 176ft[/tex]
At time t=6, the height of the ball is
[tex]f(6)=-16(6)^2+192(6)[/tex]
[tex]f(6)=-16(36)+192(6)[/tex]
[tex]f(6)=-576+1152 = 576[/tex]
The average velocity
[tex] = \frac{f(6) - f(1)}{6 - 1} [/tex]
[tex]= \frac{576- 176}{6 - 1} [/tex]
[tex]= \frac{400}{5} [/tex]
[tex] = 80 {ms}^{ - 1} [/tex]
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%.
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 testing whether the means of two normal populations are equal, summary statistics computed for two independent samples are as follows:
Brand X n2=20 xbar 2=6.80 s2=1.15
Brand Y n1=20 xbar1=7.30 s1=1.10
Assume that the population variances are equal. Then, the standard error of the sampling distribution of the sample mean difference xbar1−xbar2 is equal to: Question 2 options: (a) 1.1275 (b) 0.1266 (c) 1.2663 (d) 0.3558.
Answer: (d) 0.3558.
Step-by-step explanation:
We know that the standard error of sample mean difference is given by:-
[tex]S.E.=\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}[/tex]
Given : [tex]n_1= 20\ ,\ n_2=20[/tex]
[tex]s_1=1.10\ ,\ \ s_2=1.15[/tex]
Then , the standard error of the sampling distribution of the sample mean difference [tex]\overline{x_1}-\overline{x_2}[/tex] is equal to :-
[tex]S.E.=\sqrt{\dfrac{1.10^2}{20}+\dfrac{1.15^2}{20}}\\\\\Rightarrow\ S.E.=0.355844066973\approx0.3558[/tex]
Hence, the standard error of the sampling distribution of the sample mean difference [tex]\overline{x_1}-\overline{x_2}[/tex] is equal to 0.3558.
Final answer:
The standard error of the sampling distribution of the sample mean difference is calculated using the formula involving standard deviations and sample sizes of the independent samples; the correct answer, after computation, is 0.3558.
Explanation:
The standard error of the sampling distribution of the sample mean difference (ëxbar1 - ëxbar2) when assuming population variances are equal can be computed using the formula for the standard error of the difference of two independent sample means, which is the square root of the sum of their variances divided by their respective sample sizes. The formula is:
SE = √((s1²/n1) + (s2²/n2))
Given the summary statistics:
n1 = n2 = 20 (sample sizes)s1 = 1.10 (standard deviation of sample 1)s2 = 1.15 (standard deviation of sample 2)The calculation of the standard error would be:
SE = √((1.10²/20) + (1.15²/20))
SE = √((1.21/20) + (1.3225/20))
SE = √(0.0605 + 0.066125)
SE = √(0.126625)
SE = 0.3558 (when rounded to four decimal places)
Hence, the correct answer is option (d) 0.3558.
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.
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
The Royal Fruit Company produces two types of fruit drinks. The first type is 55% pure fruit juice, and the second type is 80% pure fruit juice. The company is attempting to produce a fruit drink that contains 65% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 80 pints of a mixture that is 65%
pure fruit juice?
Answer:
First type of fruit drinks: 48 pints
Second type of fruit drinks: 32 pints
Step-by-step explanation:
Let's call A the amount of first type of fruit drinks. 55% pure fruit juice
Let's call B the amount of second type of fruit drinks. 80% pure fruit juice
The resulting mixture should have 65% pure fruit juice and 80 pints.
Then we know that the total amount of mixture will be:
[tex]A + B = 80[/tex]
Then the total amount of pure fruit juice in the mixture will be:
[tex]0.55A + 0.8B = 0.65 * 80[/tex]
[tex]0.55A + 0.8B = 52[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.8 and add it to the second equation:
[tex]-0.8A -0.8B = -0.8*80[/tex]
[tex]-0.8A -0.8B = -64[/tex]
[tex]-0.8A -0.8B = -64[/tex]
+
[tex]0.55A + 0.8B = 52[/tex]
--------------------------------------
[tex]-0.25A = -12[/tex]
[tex]A = \frac{-12}{-0.25}[/tex]
[tex]A = 48\ pints[/tex]
We substitute the value of A into one of the two equations and solve for B.
[tex]48 + B = 80[/tex]
[tex]B = 32\ pints[/tex]
To create an 80-pint batch of 65% pure fruit juice, the Royal Fruit Company needs to solve two equations representing the volume and percent mixture of the two juices. These equations can be solved simultaneously to find the required volumes of each juice.
Explanation:The subject of this question falls under Mathematics, particularly dealing with proportions and algebra. Given that the first type of juice is 55% pure fruit and the second type is 80% pure fruit, we can define our variables: let's denote X as the volume of the first type of drink and Y as the volume of the second one. We know that the total volume is 80 pints, so we have our first equation: X + Y = 80. The second equation derives from the percentage of fruit juice: 0.55X + 0.80Y = 0.65*80.
Now we can solve these two equations to find the volumes of X and Y. The solution to these equations will provide us with the volume needed from each of the two types of juice to achieve a 65% pure fruit juice drink.
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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
...
How many millimeters are in 4.3 centimeters? How many centimeters are in 57 millimeters? Approximately how many centimeters are in an inch?
Answer:
a) 43
b) 5.7
c) 2.54
Step-by-step explanation:
The metric system is so easy to work with, everything is in base of 10, and all measures use a prefix. Here, we're talking about length, with the main measure being the meter.
1 m = 100 centimeters (centi- = 1/100)
1 m = 1000 millimeters (milli- = 1/1000)
So, for each centimeter, you have 10 millimeters.
a) How many millimeters are in 4.3 centimeters?
1 cm = 10 mm as shown above, so 4.3 cm = 43 mm
b) How many centimeters are in 57 millimeters?
10 mm = 1 cm so, 57 mm = 5.7 cm
c) Approximately how many centimeters are in an inch?
There are approximately 2.54 cm in an inch. No real calculation to make here, it's just a unit conversion between systems, done following a known reference table.
A certain group of women has a 0.640.64%
rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness?
What is the probability that the woman selected does not have red/green color blindness?
nothing
(Type an integer or a decimal. Do not round.)
Answer:
the probability that the woman selected does not have red/green color blindness is 0.9936.
Step-by-step explanation:
Final answer:
The probability that a randomly selected woman does not have red/green color blindness is 99.36%.
Explanation:
If the rate of red/green color blindness among a certain group of women is 0.64%, this means that out of every 100 women, 0.64 women on average would have red/green color blindness.
The complement of a probability event occurring is equal to 1 minus the probability of the event.
Therefore, the probability that a randomly selected woman does not have red/green color blindness is :
1 - 0.0064
which is 0.9936 or 99.36%.
A quality control inspector has drawn a sample of 1414 light bulbs from a recent production lot. If the number of defective bulbs is 22 or more, the lot fails inspection. Suppose 20%20% of the bulbs in the lot are defective. What is the probability that the lot will fail inspection? Round your answer to four decimal places.
Answer: 0.8021
Step-by-step explanation:
The given problem is a binomial distribution problem, where
[tex]n=14,\ p=0.2, q=1-0.2=0.8[/tex]
The formula of binomial distribution is :-
[tex]P(X=r)=^{n}C_{r}p^{r}q^{n-r}[/tex]
The probability that the lot will fail inspection is given by :_
[tex]P(X\geq2)=1-(P(X\leq1))\\\\=1-(P(0)+P(1))\\\\[/tex]
[tex]=1-(^{14}C_{0}(0.2)^{0}(0.8)^{14-0}+^{14}C_{1}(0.2)^{1}(0.8)^{14-1})\\\\=1-((1)(0.8)^{14}+(14)(0.2)(0.8)^{13})\\\\=0.802087907\approx0.8021[/tex]
Hence, the required probability = 0.4365
100 people responded to a survey about their ice cream preferences, and listed below are the results. 55 liked vanilla 30 liked chocolate 40 liked strawberry 10 liked both vanilla and strawberry 10 liked both strawberry and chocolate 15 liked both vanilla and chocolate 5 liked all three flavors How many did not like any of the three flavors?
To find the number of people who did not like any of the three flavors, we need to subtract the number of people who liked at least one flavor from the total number of people.
Explanation:To find the number of people who did not like any of the three flavors, we need to subtract the number of people who liked at least one flavor from the total number of people.
From the given information, we can create a Venn diagram to represent the preferences:
Picking it up from the explanation above, it becomes clear that 10 people liked both vanilla and strawberry, 10 people liked both strawberry and chocolate, and 15 people liked both vanilla and chocolate. We also know that 5 people liked all three flavors. Using this information, we can determine the number of people who liked at least one flavor by adding up the numbers in the overlapping circles: 10 + 10 + 15 + 5 = 40 people.
To find the number of people who did not like any of the three flavors, we subtract 40 from the total number of people who responded to the survey: 100 - 40 = 60 people.
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To find out how many people did not like any of the three flavors, we use the principle of inclusion-exclusion, resulting in 5 people who did not like vanilla, chocolate, or strawberry.
The student has asked a question related to combinations without repetition and the interpretation of survey results. To determine how many people did not like any of the three flavors (vanilla, chocolate, strawberry), we can use the principle of inclusion-exclusion. Here are the steps to solve this:
First, add the number of people who liked each flavor: 55 (vanilla) + 30 (chocolate) + 40 (strawberry) = 125.Next, subtract the numbers who liked each combination of two flavors: 125 - (10 + 10 + 15) = 125 - 35 = 90.Now, add back those who liked all three flavors because they were subtracted twice: 90 + 5 = 95.Finally, since there were 100 people surveyed, subtract the number who liked at least one flavor from the total: 100 - 95 = 5.Therefore, 5 people did not like any of the three flavors.
you begin playing a new game called hooville. You are King of Hooville, a city of owls that is located in the treetops near Fords of Beruna. In order to know how much food to produce each year, you must predict the population of Hooville. History shows that the population growth rate of Hooville is 3.5%. The current population of owls is 80,000. Using the monetary growth formula that you used in the Uncle Harold problem, write a new function for the population of hooville. (let n=1.) PLEASE HELP. I HAVE NO IDEA WHAT IM DOING!!
Answer:
Part 1) [tex]y=80,000(1.035)^{x}[/tex]
Part 2) The table in the attached figure
Part 3) The graph in the attached figure
Step-by-step explanation:
Part 1) Find the population function
In this problem we have a exponential function of the form
[tex]y=a(b)^{x}[/tex]
where
y ----> is the population
x ----> the time in years
a is the initial value (a=80,000 people)
b is the base (b=100%+3.5%=103.5%=1.035)
substitute
[tex]y=80,000(1.035)^{x}[/tex]
Part 2) Construct the table
For x=0 years
substitute in the function equation
[tex]y=80,000(1.035)^{0}=80,000\ people[/tex]
For x=10 years
substitute in the function equation
[tex]y=80,000(1.035)^{10}=112.848\ people[/tex]
For x=20 years
substitute in the function equation
[tex]y=80,000(1.035)^{20}=159,183\ people[/tex]
For x=40 years
substitute in the function equation
[tex]y=80,000(1.035)^{40}=316,741\ people[/tex]
For x=50 years
substitute in the function equation
[tex]y=80,000(1.035)^{50}=446,794\ people[/tex]
For x=75 years
substitute in the function equation
[tex]y=80,000(1.035)^{75}=1,055,884\ people[/tex]
For x=100 years
substitute in the function equation
[tex]y=80,000(1.035)^{100}=2,495,313\ people[/tex]
Part 3) The graph in the attached figure
A composite figure is divided into two congruent trapezoids, each with a height of 4 cm.
What is the area of this composite figure?
Answer:
The area of this composite figure is [tex]64\ cm^{2}[/tex]
Step-by-step explanation:
we know that
If the composite figure is divided into two congruent trapezoids, then the area of the composite figure is equal to the area of one trapezoid multiplied by two
so
The area of the composite figure is
[tex]A=2[\frac{1}{2}(b1+b2)h][/tex]
[tex]A=(b1+b2)h[/tex]
substitute the values
[tex]A=(6+10)4[/tex]
[tex]A=64\ cm^{2}[/tex]
Answer:
64cm
Step-by-step explanation:
I looked at the guy's answer on top of mine and it was correct, go thank him!
The inverse function of f(x) = ex has a asymptote at
Answer:
x=0
Step-by-step explanation:
You are given the function [tex]f(x)=e^x.[/tex] To find it inverse function, express x in terms of y:
[tex]y=e^x\\ \\\ln y=x[/tex]
Now change x into y and y into x:
[tex]y=\ln x\\ \\f^{-1}(x)=\ln x[/tex]
The graph of the function [tex]f^{-1}(x)[/tex] has vertical asymptote x=0.
Vertical asymptote at x = 0
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.
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]
An actor invests some money at 7%, and $24000 more than four times the amount at 8%. The total annual interest earned from the investment is $29220. How much
did he invest at each amount? Use the six-step method.
He invested $_____at 7% and ____at 8%.
Answer:
$70,000 at 7%$304,000 at 8%Step-by-step explanation:
Given:
Total interest earned on two investments is $29,220.
An amount is invested at 7%.
$24,000 more than 4 times that amount is invested at 8%.
Find:
The amount invested at each rate.
Solution:
Our strategy will be to define a variable representing the amount invested at 7%, use that variable to write an expression for the amount invested at 8%, then write an equation for the total return on the investments.
Let x represent the amount invested at 7%. Then (24,000+4x) will be the amount invested at 8%. The total interest earned will be ...
interest on 7% account + interest on 8% account = total interest
0.07x + 0.08(24000+4x) = 29220
0.39x + 1920 = 29220 . . . . . . . . . . . simplify
0.39x = 27300 . . . . . . . . . . . . . . . . . .subtract 1920
x = 27300/0.39 = 70000 . . . . . . . . . divide by the coefficient of x
24,000 +4x = 24,000 +280,000 = 304,000 . . . . amount invested at 8%
He invested $70,000 at 7% and $304,000 at 8%.
Check
The answer must satisfy ...
7% interest + 8% interest = 29,220
0.07×70,000 +0.08×304,000 = 4,900 +24,320 = 29,220 . . . . as required
_____
Comment on 6-step method
We have tried to hit the highlights. Your steps appear to be ...
Identify the given information (Given)Identify the question you are asked to answer (Find)Identify the useless information in the problem statement (is none)Decide on a strategy. Make a model or drawing. (model equation shown)Solve and show work (Solution)Explain why the answer makes sense (Check)Lester Hollar is vice president for human resources for a large manufacturing company. In recent years, he has noticed an increase in absenteeism that he thinks is related to the general health of the employees. Four years ago, in an attempt to improve the situation, he began a fitness program in which employees exercise during their lunch hour. To evaluate the program, he selected a random sample of eight participants and found the number of days each was absent in the six months before the exercise program began and in the six months following the exercise program. Below are the results. At the .05 significance level, can he conclude that the number of absences has declined? Estimate the p-value.
Final answer:
Lester Hollar can assess the impact of the fitness program on absenteeism by conducting a paired samples t-test at a 0.05 significance level. If the p-value is less than 0.05, the program effectively reduced absences; otherwise, there is insufficient evidence to conclude its effectiveness.
Explanation:
Evaluating the Exercise Program’s Impact on Employee Absenteeism
Lester Hollar wishes to assess if the company’s fitness program led to a decline in employee absences. With a sample of eight participants, he analyzed absenteeism before and after the program’s implementation. To determine if absences have decreased, a hypothesis test at the 0.05 significance level (alpha) is conducted.
To evaluate the change in absences, two sets of absenteeism data are compared using a statistical test, such as the paired samples t-test. This test examines if the mean difference in absences before and after the program is statistically significant. If the p-value obtained from the test is less than the significance level of 0.05, the null hypothesis (no change in absences) would be rejected, suggesting that the exercise program was effective in reducing absences.
If Lester Hollar finds a p-value greater than 0.05, he would not reject the null hypothesis, indicating that there isn’t sufficient evidence to conclude the program's impact. It’s also important to estimate the p-value precisely as it gives a measure of the strength of evidence against the null hypothesis. However, without the specific data, we cannot calculate the p-value or make a definitive conclusion here.
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 ..
what is m angle ABC
Answer:
3rd option: 60 degrees
Step-by-step explanation:
We can see in the diagram that the angle on C is a supplementary angle, which means that the sum of 135 and internal angle will be equal to 180 degrees.
Let x be the internal angle,
Then
x+135 = 180
x = 180-135
x = 45 degrees
So now we know that two interior angles of the triangle.
Also we know that sum of all internal angles of triangle is 180 degrees.
Using the same postulate:
A+B+C = 180
75 + B + 45 = 180
120+B = 180
B = 180 - 120
B = 60 degrees
So,
third option is the correct answer ..
Answer:
It’s 120 I got it right on the test !
(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]
The total annual spending by tourists in a resort city is $450 million. Approximately 80% of that revenue is again spent in the resort city, and of that amount approximately 80% is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated by the $450 million and find the sum of the series.
Answer:
G.P. is, 450, 360, 288, 230.4,......
The sum is 2250.
Step-by-step explanation:
Given,
The first total spending in the resort city = $ 450 million,
Also, 80% of that revenue is again spent in the resort city, and of that amount approximately 80% is again spent in the same city, and so on.
Thus, there is a G.P. that represents the given situation,
450, 360, 288, 230.4,......
Which is an infinite geometric series having first term, a = 450,
Common ratio, r = 0.8,
Hence, the sum of the series,
[tex]S_n=\frac{a}{1-r}[/tex]
[tex]=\frac{450}{1-0.8}[/tex]
[tex]=\frac{450}{0.2}[/tex]
[tex]=2250[/tex]
Please. Answer Fast! Use composition to determine if G(x) or H(x) is the inverse of F(x) for the
domain x ≥ 2.
will mark brainliest
Answer:
A. H(x) is an inverse of F(x)
Step-by-step explanation:
The given functions are:
[tex]F(x)=\sqrt{x-2}[/tex]
[tex]G(x)=(x-2)^2[/tex]
[tex]H(x)=x^2+2[/tex]
We compose F(x) and G(x) to get:
[tex](F\circ G)(x)=F(G(x))[/tex]
[tex](F\circ G)(x)=F((x-2)^2)[/tex]
[tex](F\circ G)(x)=\sqrt{(x-2)^2-2}[/tex]
[tex](F\circ G)(x)=\sqrt{x^2-4x+4-2}[/tex]
[tex](F\circ G)(x)=\sqrt{x^2-4x+2}[/tex]
[tex](F\circ G)(x)\ne x[/tex]
Hence G(x) is not an inverse of F(x).
We now compose H(x) and G(x).
[tex](F\circ H)(x)=F(H(x))[/tex]
[tex](F\circ H)(x)=F(x^2+2)[/tex]
[tex](F\circ H)(x)=\sqrt{x^2+2-2}[/tex]
We simplify to get:
[tex](F\circ H)(x)=\sqrt{x^2}[/tex]
[tex](F\circ H)(x)=x[/tex]
Since [tex](F\circ H)(x)=x[/tex], H(x) is an inverse of F(x)
Rhea is solving a math puzzle. To find the solution of the puzzle, she must find the product of two numbers. The first number is the sum of 23 and x, and the second number is 18 less than two times the first number. Which of the following functions represents the product of these two numbers?
Answer:
Function which represents the product of these two numbers is:
(23+x)(28+2x)
Step-by-step explanation:
The first number is the sum of 23 and x
i.e. First number=23+x
The second number is 18 less than two times the first number.
i.e. Second number=2(23+x)-18
= 46+2x-18
= 28+2x
Product of the two numbers=(23+x)(28+2x)
Hence, function which represents the product of these two numbers is:
(23+x)(28+2x)
The average annual salary for employees in a store is $50,000. It is given that the population standard deviation is $4,000. Suppose that a random sample of 70 employees will be selected from the population.What is the value of the standard error of the average annual salary? Round your answer to the nearest integer.
Answer: 478
Step-by-step explanation:
The formula to calculate the standard error of the population mean is given by :-
[tex]S.E.=\dfrac{\sigma}{\sqrt{n}}[/tex], where [tex]\sigma[/tex] is the standard deviation and 'n' is the sample size.
Given: Mean : [tex]\mu=$\50,000[/tex]
Standard deviation : [tex]\sigma= $\4,000[/tex]
Sample size : [tex]n=70[/tex]
Now, the value of the standard error of the average annual salary is given by :-
[tex]S.E.=\dfrac{50000}{\sqrt{70}}=478.091443734\approx478[/tex]
Hence, the standard error of the average annual salary = 478
The value of the standard error of the average annual salary is calculated using the formula SE = population standard deviation / [tex]\sqrt{sample size}[/tex] ,which for a population standard deviation of $4,000 and a sample size of 70 employees, comes out to approximately $478.
To calculate the standard error of the average annual salary, you'd use the formula for the standard error of the mean when you know the population standard deviation: SE = σ /[tex]\sqrt{n}[/tex], where σ is the population standard deviation and n is the sample size.
In this case, the population standard deviation σ is $4,000, and the sample size n is 70 employees. Using the formula, we get SE = 4000 / [tex]\sqrt{70}[/tex], which will give us the standard error of the average annual salary.
Performing the calculation: SE = 4000 / [tex]\sqrt{70}[/tex] ≈ 4000 / 8.367 = 478.29, which when rounded to the nearest integer is $478. Therefore, the value of the standard error of the average annual salary is approximately $478.