Answer:
The system has one solution.
Step-by-step explanation:
We have two equations:
y = -2x - 4
y = 3x + 3
Equalling them:
y = y
-2x - 4 = 3x + 3
5x = -7
[tex]x = -\frac{7}{5}[/tex]
And
[tex]y = 3x + 3 = 3(-\frac{7}{5}) + 3 = \frac{-21}{5} + 3 = \frac{-21}{5} + \frac{15}{5} = -\frac{6}{5}[/tex]
Replacing in the other equation we should get the same result.
[tex]y = -2x - 4 = -2(-\frac{7}{5}) - 4 = \frac{14}[5} - 4 = \frac{14}{4} - \frac{20}{5} = -\frac{6}{5}[/tex]
So the system has one solution.
Please help. I’ll mark you as brainliest if correct!
Answer:
[tex]\sqrt{-6} \sqrt{-384}=\sqrt{(-6)(-384)}=\sqrt{2304}=48\\ a=48\\b=0[/tex]
or
[tex]a=-48\\b=0[/tex]
both solutions are correct because root square has two solutions, one positive and one negative.
Answer:
a= -48
b=0
Step-by-step explanation:
[tex]\sqrt[]{-6} = i\sqrt{6}[/tex]
[tex]\sqrt{-384} =i\sqrt{384}[/tex]
[tex](i\sqrt{6} )(i\sqrt{384} )[/tex]
[tex]i^{2} \sqrt{2304}[/tex]
(-1)(48) = -48
a + bi
a= -48
b= 0
The method of Problem 20 can be extended to second order equations with variable coefficients. If y1 is a known nonvanishing solution of y′′ + p(t)y′ + q(t)y = 0, show that a second solution y2 satisfies (y2 /y1 )′ = W (y1 , y2 )/y21 , where W (y1 , y2 ) is the Wronskian of y1 and y2 . Then use Abel’s formula [Eq. (23) of Section 3.2] to determine y2 .
Answer:24y
Step-by-step explanation:
3/5 of a juice drink is made of real juice. What percent of the drink is
real juice?
Answer:
60%
Step-by-step explanation:
Percent means out of 100
Changing 3/5 to a denominator of 100
3/5*20/20
60/100
The percent is 60 %
The mean annual salary for intermediate level executives is about $74000 per year with a standard deviation of $2500. A random sample of 36 intermediate level executives is selected. What is the probability that the mean annual salary of the sample is between $71000 and $73500?
Answer:
11.51% probability that the mean annual salary of the sample is between $71000 and $73500
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
[tex]\mu = 74000, \sigma = 2500, n = 36, s = \frac{2500}{\sqrt{36}} = 416.67[/tex]
What is the probability that the mean annual salary of the sample is between $71000 and $73500?
This is the pvalue of Z when X = 73500 subtracted by the pvalue of Z when X = 71000. So
X = 73500
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{73500 - 74000}{416.67}[/tex]
[tex]Z = -1.2[/tex]
[tex]Z = -1.2[/tex] has a pvalue of 0.1151
X = 71000
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{71000 - 74000}{416.67}[/tex]
[tex]Z = -7.2[/tex]
[tex]Z = -7.2[/tex] has a pvalue of 0.
0.1151 - 0 = 0.1151
11.51% probability that the mean annual salary of the sample is between $71000 and $73500
22,056 people went to the baseball game on Sunday. Half as many people came on money. How many people were at the baseball game on Sunday and Monday altogether?
Answer:33084
Step-by-step explanation:
22056 divided by 2 = Monday
Monday= 11028
11028+22056=33084
Answer:
33084 People were at the baseball game on Sunday and ~Money~ Monday all together.
Step-by-step explanation:
Sunday - 22056
Monday - "Half as many" 22056 Divided by 2
= 11028
Altogether - Sunday + Monday = 33084
As a shortcut on your calculator, you could do:
22056 + (22056 divided by 2)
= 33084
Classify the following triangle .check all that apply
Answer:
acute and scalene
Step-by-step explanation:
Answer:no entiendo esta en ingles
Step-by-step explanation:
what function is increasing? will give brainlist !
Answer:
Option B.
Step-by-step explanation:
Option A.
f(x) = [tex](0.5)^{x}[/tex]
Derivative of the given function,
f'(x) = [tex]\frac{d}{dx}(0.5)^x[/tex]
= [tex](0.5)^x[\text{ln}(0.5)][/tex]
= [tex]-(0.693)(0.5)^{x}[/tex]
Since derivative of the function is negative, the given function is decreasing.
Option B. f(x) = [tex]5^x[/tex]
f'(x) = [tex]\frac{d}{dx}(5)^x[/tex]
= [tex](5)^x[\text{ln}(5)][/tex]
= [tex]1.609(5)^x[/tex]
Since derivative is positive, given function is increasing.
Option C. f(x) = [tex](\frac{1}{5})^x[/tex]
f'(x) = [tex]\frac{d}{dx}(\frac{1}{5})^x[/tex]
= [tex]\frac{d}{dx}(5)^{(-x)}[/tex]
= [tex]-5^{-x}.\text{ln}(5)[/tex]
Since derivative is negative, given function is decreasing.
Option D. f(x) = [tex](\frac{1}{15})^x[/tex]
f'(x) = [tex]-15^{-x}[\text{ln}(15)][/tex]
= [tex]-2.708(15)^{-x}[/tex]
Since derivative is negative, given function is decreasing.
Option (B) is the answer.
A diagonal of a cube measures 30 inches. The diagonal of a face measures StartRoot 600 EndRoot inches.
In inches, what is the length of an edge of the cube? Round the answer to the nearest tenth.
Answer:
17.3 Inches
Step-by-step explanation:
Given that the diagonal of a cube = 30 inches
For a cube of side length s, Length of its diagonal [tex]=s\sqrt{3}[/tex]
Therefore:
[tex]s\sqrt{3}=30\\$Divide both sides by \sqrt{3}\\s=30 \div \sqrt{3}\\s=17.3$ inches (to the nearest tenth.)[/tex]
Side Length of the cube is 17.3 Inches.
Answer:
17.3
Step-by-step explanation:
Edge 2020
A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that when this service life is in control it is normally distributed with a mean of 500 hours and a standard deviation of 20 hours. On three recent production batches, he tested service life on random samples of four headlamps, with these results: Sample Service Life (hours) 1 495 500 505 500 2 525 515 505 515 3 470 480 460 470 What is the mean of the sampling distribution of sample means when the service life is in control
Answer:
[tex]$ \text {Sample mean} = \bar{x} = \mu = 500 \: hours $[/tex]
Step-by-step explanation:
What is Normal Distribution?
We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.
For the given scenario, it is known from numerous previous samples that when this service life is in control it is normally distributed with a mean of 500 hours and a standard deviation of 20 hours.
On three recent production batches, he tested service life on random samples of four headlamps.
We are asked to find the mean of the sampling distribution of sample means when the service life is in control.
Since we know that the population is normally distributed and a random sample is taken from the population then the mean of the sampling distribution of sample means would be equal to the population mean that is 500 hours.
[tex]$ \text {Sample mean} = \bar{x} = \mu = 500 \: hours $[/tex]
Whereas the standard deviation of the sampling distribution of sample means would be
[tex]\text {standard deviation} = s = \frac{\sigma}{\sqrt{n} } \\\\[/tex]
Where n is the sample size and σ is the population standard deviation.
[tex]\text {standard deviation} = s = \frac{20}{\sqrt{4} } \\\\ \text {standard deviation} = s = \frac{20}{2 } \\\\ \text {standard deviation} = s = 10 \: hours \\\\[/tex]
Solve the equation and state a reason for each step.
23+11a-2c=12-2c
Simplifying
23 + 11a + -2c = 12 + -2c
Add '2c' to each side of the equation.
23 + 11a + -2c + 2c = 12 + -2c + 2c
Combine like terms: -2c + 2c = 0
23 + 11a + 0 = 12 + -2c + 2c
23 + 11a = 12 + -2c + 2c
Combine like terms: -2c + 2c = 0
23 + 11a = 12 + 0
23 + 11a = 12
Solving
23 + 11a = 12
Solving for variable 'a'.
Move all terms containing a to the left, all other terms to the right.
Add '-23' to each side of the equation.
23 + -23 + 11a = 12 + -23
Combine like terms: 23 + -23 = 0
0 + 11a = 12 + -23
11a = 12 + -23
Combine like terms: 12 + -23 = -11
11a = -11
Divide each side by '11'.
a = -1
Simplifying
a = -1
Carefully review the research matrix presented below. If this is a within subjects design, how many total participants will be used in the experiment?
Immaculate Appearance Neat Appearance Sloppy Appearance
15 participants 15 participants 15
participants
a. 15
b. 30
c. 45
d. 60
Answer:
c. 45
Step-by-step explanation:
there are 15 participant in each category, and there are 3 categories, so total participants = 15 * 3
= 45
Hope this helps, and please mark me brainliest if it does!
In the first year if ownership, a new car lose 20% of its value. If a car lost $4,200 value in the first year, how much did the car originally cost?
Answer:
21,000$
Step-by-step explanation:
part to whole method
20/100 and 4,200/
How many 20s to get to 4,200?
Scientists think that robots will play a crucial role in factories in the next several decades. Suppose that in an experiment to determine whether the use of robots to weave computer cables is feasible, a robot was used to assemble 507 cables. The cables were examined and there were 9 defectives. If human assemblers have a defect rate of 0.035 (3.5%), does this data support the hypothesis that the proportion of defectives is lower for robots than humans
Answer:
The data support the hypothesis that the proportion of defectives is lower for robots than humans.
Step-by-step explanation:
To know if the proportion of defectives is lower for robots than humans so as to prove if the hypothesis is true.
From the data given:
Total number of cables a robot assembled = 507
Defectives = 9
To get the defect rate = the number of defects divided by the total number of cables, multiplied by 100.
Defect rate = (9 / 507) x 100 = 0.01775 x 100
Defect rate for the robot = 1.775%
From the question, a robot was used and the defect rate after the calculation is 1.775%. While for humans, the defect rate is 3.5%. This implies, if humans were used to assembling the same 507 cables, there will be 17.745 defectives.
x / 507 = 3.5%
x (defectives) = 17.745
Therefore, the data support the hypothesis that the proportion of defectives is lower for robots than humans.
The hypotenuse of a right triangle is 95 inches long. One leg is 4 inch(es) longer than the other. Find the lengths of the legs of the triangle. Round your answers to the nearest tenth of an inch.
Answer:
65.1 and 69.1
Step-by-step explanation:
a^2+b^2=c^2
c=95
b=a+4
Solve for a^2+(a+4)^2=95^2
a=65.1
b=a+4=69.1
Answer:
65.1 and 69.1
Step-by-step explanation:
c² = a² + b²
c= 95
a - one leg
b= (a + 4) - second leg
95² = a² + (a + 4)²
9025 = a² + a² + 2*4a + 16
2a² + 8a - 9009 = 0
[tex]a= \frac{-b +/-\sqrt{b^2 - 4ac} }{2a} \\\\a = \frac{-8 +/-\sqrt{8^2 - 4*2*9009} }{2*2} \\\\a=65.1 \ and \ a=- 69.1[/tex]
A leg length can be only positive. a = 65.1
b = 65.1 + 4 = 69.1
Find the x- and y-intercepts of the equation 7x + 14y = 28.
Answer: The x-intercept is 4 and the y-intercept is 2.
Step-by-step explanation:
The x is intercept is when y is 0 and the y intercept is when x is 0.So using this information you can put in 0 for x and another 0 for y and solve for the x and y intercepts.
7(0) + 14y = 28
0 + 14y = 28
14y = 28
y = 2
7x + 14(0) = 28
7x + 0 = 28
7x = 28
x = 4
The [tex]x[/tex]-intercept of the given equation is [tex]4[/tex] and the [tex]y[/tex]-intercept is [tex]2[/tex].
Given:
The equation is:
[tex]7x+14y=28[/tex]
To find:
The [tex]x[/tex]-intercept and [tex]y[/tex]-intercept of the given equation.
Explanation:
We have,
[tex]7x+14y=28[/tex] ...(i)
Substitute [tex]x=0[/tex] in (i) to find the [tex]y[/tex]-intercept.
[tex]7(0)+14y=28[/tex]
[tex]14y=28[/tex]
[tex]\dfrac{14y}{14}=\dfrac{28}{14}[/tex]
[tex]y=2[/tex]
Substitute [tex]y=0[/tex] in (i) to find the [tex]x[/tex]-intercept.
[tex]7x+14(0)=28[/tex]
[tex]7x=28[/tex]
[tex]\dfrac{7x}{7}=\dfrac{28}{7}[/tex]
[tex]x=4[/tex]
Therefore, the [tex]x[/tex]-intercept of the given equation is [tex]4[/tex] and the [tex]y[/tex]-intercept is [tex]2[/tex].
Learn more:
https://brainly.com/question/19669786
Given the following diagram, are and opposite rays? yes no
Answer:
Where is the diagram?
Step-by-step explanation:
Answer:
OC and OE are apposite rays so the Answer is yes
What letter completes the puzzle? The answer is probably easy for you guys but I don't understand how the letters go along with the puzzle. Thank you!
Answer:
the answer is E the number at the top tells you which position it falls under in the alphabet
A tree casts an 8-foot shadow on the ground. The length from the tip of the shadow to the top of the tree is 17 feet. What is the height of the tree?
Answer:
Height of tree = 15 ft
Step-by-step explanation:
Given:
Length of shadow (Base) = 8 ft
Length from the tip to top of the tree (Hypotenues) = 17 ft
Find:
Height of tree = ?
Computation:
Using Pythagoras theorem:
[tex]Height\ of\ tree = \sqrt{Hypotenues^2 - base^2} \\\\Height\ of\ tree = \sqrt{17^2 - 8^2} \\\\Height\ of\ tree = \sqrt{289-64}\\\\Height\ of\ tree = \sqrt{225}\\\\ Height\ of\ tree =15[/tex]
Height of tree = 15 ft
Answer:
The answer is 15 feet from the ground to the top of the tree.
Step-by-step explanation:
A farmer was interest in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor
Answer:
= 0.0041
Step-by-step explanation:
Given that:
A farmer was interest in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away
mean number of flights to be 57
a standard deviation of 12
fewer flights on average in the next 40 rows
[tex]\mu = 57\\\\\sigma=12\\\\n=40[/tex]
so,
[tex]P(x<52)[/tex]
[tex]=P(\frac{x-\mu}{\sigma/\sqrt{n} } <\frac{52-57}{12/\sqrt{40} } )\\\\=P(z<\frac{-5\times6.325}{12} )\\\\=P(z<\frac{-31.625}{12})\\\\=P(z<-2.64)[/tex]
using z table
= 0.0041
The probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor is 0.0041 and this can be determined by using the properties of probability.
Given :
The distribution of grasshoppers may not be normally distributed in his field due to growing conditions.The mean number of flights to be 57 with a standard deviation of 12.The probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor, can be determined by using the following calculations:
[tex]\rm P(x<52)=P\left (\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n} }}<\dfrac{52-57}{\dfrac{12}{\sqrt{40} }}\right)[/tex]
[tex]\rm P(x<52)=P\left (z<\dfrac{-5\times 6.325}{12 }}\right)[/tex]
[tex]\rm P(x<52)=P\left (z<\dfrac{-31.625}{12 }}\right)[/tex]
[tex]\rm P(x<52)=P\left (z<-2.64\right)[/tex]
Now, using z-table:
P(x < 52) = 0.0041
For more information, refer to the link given below:
https://brainly.com/question/21586810
Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow.
Market
Weekly Gross Revenue
($100s)
Television Advertising
($100s)
Newspaper Advertising
($100s)
Mobile
101.3
5
1.5
Shreveport
51.9
3
3
Jackson
74.8
4
1.5
Birmingham
126.2
4.3
4.3
Little Rock
137.8
3.6
4
Biloxi
101.4
3.5
2.3
New Orleans
237.8
5
8.4
Baton Rouge
219.6
6.9
5.8
(a) Use the data to develop an estimated regression with the amount of television advertising as the independent variable.
Let x represent the amount of television advertising.
If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
= + x
Test for a significant relationship between television advertising and weekly gross revenue at the 0.05 level of significance. What is the interpretation of this relationship?
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
(b) How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain?
If required, round your answer to two decimal places.
%
(c) Use the data to develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables.
Let x1 represent the amount of television advertising.
Let x2 represent the amount of newspaper advertising.
If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
= + x1 + x2
Is the overall regression statistically significant at the 0.05 level of significance? If so, then test whether each of the regression parameters β0, β1, and β2 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
(d) How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain?
If required, round your answer to two decimal places.
%
(e) Given the results in part (a) and part (c), what should your next step be? Explain.
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
(f) What are the managerial implications of these results?
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
Order the numbers from least to greatest based on their absolute values.
|23|, |−37|, |−6|, |18|, |−24|, |2|
Answer:
/-37/, /-24/, /-6/, /2/, /18/, /23/
Really in need of help :( please !
Answer:
A
Step-by-step explanation:
(2,1) - only one in the table. Remember (x,y)
What’s the correct answer for this?
Answer:
B.
Step-by-step explanation:
Since two diameters are intersecting eachother, the angles inside them would be vertical angles so they'll be congruent.
So
m<LYM = m<JYM
Also their arcs would be equal to their angles measures so,
Arc JK = 52°
Jose predicted that he would sell 48 umbrellas. He actually sold 72 umbrellas.What are the values of a and b in the table below. Round to the nearest tenth if necessary
Answer:
The answer is A
Step-by-step explanation:
Let f(x) = -2x + 7 and g(x) = -6x + 3. Find fg and state its domain.
Answer:
f(g(x))=12x+1
Step-by-step explanation:
f(g(x)) = -2(-6x+3)+7
f(g(x))= 12x-6+7
f(g(x))=12x+1
Domain: All real numbers
A public relations firm found that only 27% of voters in a certain state are satisfied with their U.S. senators. How large a sample of voters should be drawn so that the sample proportion of voters who are satisfied with their senators is approximately normally distributed?a) 38b) 14c) 10d) 48
Answer:
a) 38
Step-by-step explanation:
The normal distribution can be applied if:
[tex]np \geq 5[/tex] and [tex]n(1-p) \geq 5[/tex]
In this question:
[tex]p = 0.27[/tex]
Then
a) 38
n = 38.
Then
38*0.27 = 10.26
38*0.73 = 27.74
Satisfies. But is this the smallest sample of the options which satisfies.
b) 14
n = 14
Then
14*0.23 = 3.22
14*0.77 = 10.78
Does not satisfy
c) 10
Smaller than 14, which also does not satisfy, so 10 does not satisfy.
d) 48
Greater than 38, which already satisfies. So the answer is a)
What is the value of d21+d22+d23 given the matrix equation below?
Answer:
B. 8
Step-by-step explanation:
The question lacks the required diagram. Find the diagram in the attachment.
Before we can find d21, d22 and d23, we need to get the matrix D first as shown in the attached solution.
On comparison as shown in the attachment, d21 = 11, d22 = -10 and d23 = 7
Note that d21 refers to element in the second row and first column of the matrix
d22 is the element in the second row and second column of the matrix
d23 is the element in the second row and third column of the matrix
d21+d22+d23 = 11-10+7
d21+d22+d23 = 8
The second option is correct.
The total energy need during pregnancy is normally distributed, with a mean of 2600 kcal/day and a standard deviation of 50 kcal/day. Include your Normal curve for all parts! a) [4 pts] If one pregnancy is randomly selected, find the probability that the total energy need is more than 2650 kcal/day. b) [4 pts] The middle 30% of total energy need during pregnancy are between what values? c) [4 pts] What is the probability that a random sample of 20 pregnant women has a mean energy need of more than 2625 kcal/day?
Answer:
a) 0.3085
b) 2574
c) 0.0125
Step-by-step explanation:
mean (μ) = 2600 kcal/day and a standard deviation (σ) = 50 kcal/day
a) The z score is given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{x-\mu}{\sigma}=\frac{2650-2600}{50}=1[/tex]
From the normal distribution table, P(x > 2650) = P(z > 1) = 1 - P(z < 1) = 1 - 0.8413 = 0.1587
b) A probability of 30% corresponds with a z score of -0.52
[tex]z=\frac{x-\mu}{\sigma}\\-0.52=\frac{x-2600}{50} \\x-2600=-26\\x=2600-26\\x=2574[/tex]
c) For a sampling distribution of sample mean, the standard deviation is [tex]\frac{\sigma}{\sqrt{n} }[/tex]
The z score is given by:
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n} }}[/tex]
n = 20
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n} }}=\frac{2625-2600}{\frac{50}{\sqrt{20} }}=2.24[/tex]
From the normal distribution table, P(x > 2625) = P(z > 2.24) = 1 - P(z < 2.24) = 1 - 0.9875 = 0.0125
The tallest church tower in the Netherlands is the Dom Tower in Utrecht. If the angle of elevation to the top of the tower is 77° when 25.9 m from the base, what is the height of the Dom Tower to the nearest metre.
Answer:
Height of the Dom is 112.18 m.
Step-by-step explanation:
The tallest church tower in the Netherlands is the Dom Tower in Utrecht. The angle of elevation to the top of the tower is 77° when 25.9 m from the base. It is required to find the height of the Dom Tower. Let its height is h. So, using trigonometric formula to find it as :
[tex]\tan\theta=\dfrac{h}{b}\\\\\tan(77)=\dfrac{h}{25.9}\\\\h=\tan(77)\times 25.9\\\\h=112.18\ m[/tex]
So, the height of the Dom is 112.18 m.
A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C = C(t) of the glucose solution in the bloodstream is dC/dt = r - kC where k is a positive constant. Assuming that C0 < r/k, find lim t→[infinity] C(t) and interpret your answer
Answer:
[tex]C(t) =\dfrac{ r}{k} - \left (\dfrac{r-kC_{0}}{k} \right )e^{ -kt}[/tex]
[tex]C(t) =\dfrac{ r}{k}- e^{ -kt}[/tex] we can conclude that the function is an increasing function.
Step-by-step explanation:
Given that:
[tex]\dfrac{dC}{dt}= r-kC[/tex]
[tex]\dfrac{dC}{r-kC}= dt[/tex]
By taking integration on both sides ;
[tex]\int\limits\dfrac{dC}{r-kC}= \int\limits \ dt[/tex]
[tex]- \dfrac{1}{k}In (r-kC)= t +D[/tex]
[tex]In(r-kC) = -kt - kD \\ \\ r- kC = e^{-kt - kD} \\ \\ r- kC = e^{-kt} e^{ - kD} \\ \\r- kC = Ae^{-kt} \\ \\ kC = r - Ae^{-kt} \\ \\ C = \dfrac{r}{k} - \dfrac{A}{k}e ^{-kt} \\ \\[/tex]
[tex]C(t) =\frac{ r}{k} - \frac{A}{k}e^{ -kt}[/tex]
where;
A is an integration constant
In order to determine A, we have C(0) = C0
[tex]C(0) =\frac{ r}{k} - \frac{A}{k}e^{0}[/tex]
[tex]C_0 =\frac{r}{k}- \frac{A}{k}[/tex]
[tex]C_{0} =\frac{ r-A}{k}[/tex]
[tex]kC_{0} =r-A[/tex]
[tex]A =r-kC_{0}[/tex]
Thus:
[tex]C(t) =\dfrac{ r}{k} - \left (\dfrac{r-kC_{0}}{k} \right )e^{ -kt}[/tex]
b ) Assuming that C0 < r/k, find lim t→[infinity] C(t) and interpret your answer
[tex]C_{0} < \lim_{t \to \infty }C(t)[/tex]
[tex]C_0 < \dfrac{r}{k}[/tex]
[tex]kC_0 <r[/tex]
The equation for C(t) can therefore be re-written as :
[tex]C(t) =\dfrac{ r}{k} - \left (\dfrac{r-kC_{0}}{k} \right )e^{ -kt}[/tex]
[tex]C(t) =\dfrac{ r}{k} - \left (+ve \right )e^{ -kt} \\ \\C(t) =\dfrac{ r}{k}- e^{ -kt}[/tex]
Thus; we can conclude that the above function is an increasing function.
