Answer:
420 Square Units
Step-by-step explanation:
7*12*5 = 420 units
The average score Josie had in 6 subjects is 72 and her average score after 2 additional subjects were added is 74.25. If she scored 80 in the 7th subject, what was her score in the 8th subject correct to the nearest whole number?
Answer: 82
Step-by-step explanation:
72 times 6 is equal to 432
6 plus 2 is 8
74.25 times 8 is equal to 594
If you minus 594 and 432 you get 162
162 is total marks for both subjects
162 minus 80 you get 82 which is the marks for the 8th subject
Hope it taught you something..
00:00
Gavin counted the number of days until the end of school.
If he counted the days in groups of 7, which list shows the numbers Gavin could have
named?
O
A. 7, 15, 22, 30
B. 7, 14, 21, 30
O
C. 7, 14, 21, 28
D. 14, 21, 32, 38
Answer:
C. 7, 14, 21, 28
Step-by-step explanation:
We want to find the multiples of 7
7*1= 7
7*2 = 14
7*3= 21
7*4 = 28
Answer:
The answer is, C. 7, 14, 21, 28
A party for 325 people is booked. The cost of the party is $8,125. If the party host are given an 8% discount on their total bill, what is the cost of the party
8
.
Solve simultaneously the equations 6p - 4r = 14 and p-2r = 5.
p=..
فن
199110
Answer:
p = 1 and r = - 2
Step-by-step explanation:
Given the 2 equations
6p - 4r = 14 → (1)
p - 2r = 5 → (2)
Multiplying (2) by - 6 and adding to (1) will eliminate the term in p
- 6p + 12r = - 30 → (3)
Add (1) and (3) term by term to eliminate p
8r = - 16 ( divide both sides by 8 )
r = - 2
Substitute r = - 2 into either of the 2 equations and solve for p
Substituting into (2)
p - 2(- 2) = 5
p + 4 = 5 ( subtract 4 from both sides )
p = 1
Question 1 of 25
2 Points
What can you say about the end behavior of the function f(x) = - In (2x) + 4 ?
A. Both ends decrease
B. Both ends increase
c. One end decreases and one end approaches a constant
D. One end increases and one end decreases
SHIRT
Answer:
D. One end increases and one end decreases
One end increases and one end decreases in the given function f(x) = - In (2x) + 4. Therefore, the correct option is option D.
What is mathematical function?
In mathematics, a function is a phrase, regulation, or statute that establishes the connection between an independent variable and a dependent variable.
In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences. The German mathematician Peter Dirichlet initially offered the contemporary definition of function in 1837. One end increases and one end decreases in the given function f(x) = - In (2x) + 4.
Therefore, the correct option is option D.
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Find the area of a rectangle with side lengths 58 ft and 13 ft.
Use the drawing to help you solve the problem.
Square with side as 1 foot is divided into 24 equal rectangles.
A. 511ft2
B. 11112 ft2
C. 611ft2
D. 524ft2
Answer:
A =754 ft^2
Step-by-step explanation:
The area of a rectangle is given by
A = l*w
A = 58*13
A =754 ft^2
what is the solution of square root 2x+4=16?
A.x=6
B.x=72
C.x=126
D.no solution
Answer:A.6
Step-by-step explanation:first you subtract -4 with +4 and same ro the other side 16-4 =12. Then tha t leaves you with 2x=12 , so then you do 12÷2=6 . so x=6
The ratio of surface areas of two similar polyhedra is equal to the cube of the
ratio between their corresponding edge lengths.
O A. True
O B. False
Answer:
The answer is false already got this question wrong though D;
Step-by-step explanation:
The ratio of the volumes of two similar solid polyhedra is equal to the cube of the ratio between their edges is false.
The ratio of the corresponding edges or sides or lengths of given similar solids is called the linear scale factor.
What is the area of scale factor?
The linear scale factor can be used to determine the area scale factor which is the ratio of the area of the similar solids.
The area scale factor is given as the square of the linear scale factor.
that is Area Scale Factor = (Linear Scale Factor)²
The linear scale factor can also be used to find the ratio of the volumes of similar solids, or what is called the volume scale factor such that the volume scale factor is given by the cube of the area scale factor.
Therefore the given statement is false
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Andrea plans to serve orange juice for a large family brunch. She combines the orange concentrate and the water using a 2:5 ratio. How many cans of water does she combine with 10 cans of concentrate a 4 b 7 c 14 d 25
Answer:
d 25
So 10 cans of orange Concentrate will be = 25 cans of water
Step-by-step explanation:
Andrea combines the orange concentrate and the water using a 2:5 ratio.
We are to find how many cans of water she needs to combine with 10 cans of concentrate
So using the ratio.
2 cans of orange Concentrate is to 5 cans of water.
So 10 cans of orange Concentrate will be =( 10*5)/2
So 10 cans of orange Concentrate will be = 50/2
So 10 cans of orange Concentrate will be = 25 cans of water
write -4i+(1/4-5i)-(-3/4+8i)+17i as a complex number in the standard for
Answer:
[tex]1+ 0i[/tex]
Step-by-step explanation:
A complex number is a number which has some real part and some imaginary part.
Standard form of a complex number is represented as
[tex]a +bi[/tex]
Where [tex]a[/tex] is the real part,
and [tex]bi[/tex] is the imaginary part.
And [tex]i = \sqrt{-1}[/tex]
Given complex number:
[tex]-4i+\dfrac{1}{4}-5i)-(-\dfrac{3}{4}+8i)+17i\\\Rightarrow -4i+\dfrac{1}{4}-5i + \dfrac{3}{4}-8i+17i\\\Rightarrow \dfrac{3}{4}+\dfrac{1}{4}-4i -5i-8i+17i\\\Rightarrow \dfrac{3+1}{4}-17i+17i\\\Rightarrow \dfrac{4}{4}+0i\\\Rightarrow 1 + 0i[/tex]
Hence, the standard form is [tex]1+ 0i[/tex].
The average number of road accidents that occur on a particular stretch of road
during a year is 11.
Ake
P(x = k) =
k!
k is the given number of event occurrences
A is the average rate of event occurrences
Using the Poisson distribution formula, what is the probability of observing exactly
7 accidents on this stretch of road next year? Answer choices are rounded to the
hundredths place.
Answer:0.06
Step-by-step explanation:
(1 point) The intensity of light at a depth of x meters below the surface of a lake satisfies the differential equation dIdx=(−1.21)I. (a) At what depth, in meters, is the intensity of light half that of L, where L equals the intensity of light at the surface (where x=0)?
Answer:
0.573 m
Step-by-step explanation:
a. To find the depth, x, we first solve the differential equation to find the expression for I
dI/dx = (-1.21)I
dI = (-1.21)Idx
dI/I = -1.21dx
Integrating both sides, we have
∫dI/I = ∫-1.21dx
㏑I = -1.21x + C
I = exp(-1.21x + C)
I = exp(-1.21x)exp(C) Let exp(C) = A
I =Aexp(-1.21x)
when x = 0, I = L. Substituting these into the equation, we have
L = Aexp(-1.21 × 0)
L = Aexp(0)
L = A
So, I = Lexp(-1.21x)
we want to find x when I = L/2.
So, L/2 = Lexp(-1.21x)
1/2 = exp(-1.21x)
-1.21x= ㏑(1/2)
-1.21x= -㏑2
x = -㏑2/-1.21
x = 0.693/1.21
x = 0.573 m
A drink is a mixture of three parts water to 1 part orange concentrate. How much water should be added to 100ml of orange concentrate? *
Answer:
300 mL
Step-by-step explanation:
If one part is 100 mL, then 3 parts will be 300 mL.
300 mL of water should be added.
PLEASE HELP
Write an equation in slope-intercept form with a slope of -5 and a y-intercept of 3.
a. y = 5x + 3
C. y = 3x - 5
b. y = -5x-3
d. y = -5x+3
Answer:
d y = -5x+3
Step-by-step explanation:
slope intercept is y=mx+b m is for the slope and b is the y intercept so the -5x would go first and then the 3
Frogs have been breeding like flies at the Enormous State University (ESU) campus! Each year, the pledge class of the Epsilon Delta fraternity is instructed to tag all the frogs residing on the ESU campus. Two years ago (t = 0) they managed to tag all 46,000 of them (with little Epsilon Delta Fraternity tags). This year's pledge class discovered that all the tags had all fallen off, and they wound up tagging a total of 55,200 frogs.a) Find an exponential model for the frog population.b) Assuming exponential population growth, and that all this year's tags have fallen off, how many tags should Epsilon Delta order for next year's pledge class?
Answer:
Step-by-step explanation:
Part a
Exponential growth model for the frog population is given by
[tex]P=P_{0}b^{t}\texttt{ \ where} \ P_{0}=Population \ at \ t=0 \ and \ b=Base[/tex]
The population of frog at t=0 (2 years ago) is given as 46000 which implies
[tex]P_{0}=46000[/tex]
The population in the current year is given as 55200 which implies
At,
t=2 years, P=55200
Therefore, substitute the above values to calculate the value of b
[tex]55200=46000b^{2} => b^2=\frac{55200}{46000}=1.2[/tex]
[tex]b=\sqrt{1.2}=1.095445[/tex]
Therefore, the model equation becomes
[tex]P=46000(1.095445)^{t}[/tex]
Part b
Since all the tags for this year have fallen off which implies no of tags for the next year can be calculated by calculating the population at t=3
[tex]P=46000(1.095445)^{3}\\\\=46000\times 1.3145\\\\=60467[/tex]
Simplify m - {n + [p - (m + n - p)]}.
Answer:
Step-by-step explanation:
m-(n+(p-(m+n-p)))
Final Result :
2 • (m - p)
In a study of its reservation patterns, a national hotel chain randomly sampled the records of 20 of its locations and recorded the number of days last year when the locations had no rooms available: 163, 126, 59, 47, 146, 64, 39, 75, 86, 114, 51, 68, 58, 38 66, 72, 42, 85, 40, 59
a) Construct a histogram of these data
b) Describe the distribution
Answer:
Step-by-step explanation:
a)
Class limits Frequency
38 - 63 9
64 - 89 7
90 - 115 1
116 - 141 1
142 - 167 2
Construct a histogram of these data(image attached)
b)
Frequency distribution table (also known as frequency table) consists of various components.
Classes: A large number of observations varying in a wide range are usually classified in several groups according to the size of their values. Each of these groups is defined by an interval called class interval. The class interval between 10 and 20 is defined as 10-20.
Class limits: The smallest and largest possible values in each class of a frequency distribution table are known as class limits. For the class 10-20, the class limits are 10 and 20. 10 is called the lower class limit and 20 is called the upper class limit.
Class limit: Class limit is the midmost value of the class interval. It is also known as the mid value.
Mid value of each class = (lower limit + Upper limit)2.
If the class is 38 to 63 , lower limit is 38 and upper limit is 63. So the mid value is
[tex]\frac{(38+63)}{2} =50.5[/tex]
Class width is calculated using upper and lower value
Class width = (upper - lower) value /class
[tex]=\frac{ ( 163 - 38)}{5} \\\\=25[/tex]
Magnitude of a class interval: The difference between the upper and lower limit of a class is called the magnitude of a class interval.
Class frequency: The number of observation falling within a class interval is called class frequency of that class interval.
For which of the following numbers doesn’t the 5 appear in the hundredths place? (Multiple answers) A. 152.76 B. 26.578 C 12.057 D. 100.25 E. 253.47
Answer:
the only one that does is C
Step-by-step explanation:
doesn't= A, B, D, E
I need some help finding the answer please help
Answer:
(1,3) (B)
Step-by-step explanation:
A function cannot have 2 y values per x value. The x value of 1 has 2 y values on it which are -2 and 3. Therefore, you must remove one of them. The only one that applies is (1,3)
I completed the vertical line test which shows that A, C, nor D had another point on their lines. So B is the answer.
What is the rvalue of the following data, to three decimal places?
Answer:
b
Step-by-step explanation:
Professor Melendez has 10 students in her college algebra class. Their ages are shown below.
19, 27, 19, 18, 18, 18, 47, 19, 20, 18
The mean age of the students in the class is 22.3. The median age of the students is 19.
There are two outliers in the data from Professor Melendez’s class: 27 and 47.
When the outliers are removed, what is the mean age of the remaining students? Round your answer to the nearest tenth
What is the median age of the remaining students?
Answer:
Mean: 18.6 years old
Median: 18.5 years old
Step-by-step explanation:
Mean = average (add all numbers then divide by the number of numbers in the data set)
To find the mean, start by adding all of the numbers except for the outliers.
19 + 19 + 19 + 18 + 18 + 18 + 18 + 20 = 149
We added 8 numbers, so divide the sum by 8.
149/8 = 18.6
Therefore, the mean without outliers is 18.6 years old.
Median = middle (order numbers from least to greatest, then find the middle)
To find the median, order the numbers from least to greatest.
18, 18, 18, 18, 19, 19, 19, 20
Now, find the middle.
(Since there are an even number of numbers, select the two numbers in the middle.)
18, 18, 18, 18, 19, 19, 19, 20
Now, add these numbers and divide by 2.
18 + 19 = 37
37/2 = 18.5
Therefore, the median without outliers is 18.5 years old.
The chamber of commerce of a Florida Gulf Coast community advertises that area residential property is available at a mean cost of $125,000 or less per lot. Suppose a sample of 32 properties provided a sample mean of $130,000 per lot and a sample standard deviation of $12,500.
(a) Use a 0.05 level of significance to test the validity of the advertising claim.
(b) State the hypotheses.
(c) What is the p-value?
(d) What is your conclusion (use a level of significance of 0.05)?
(e) What is the interpretation of your conclusion?
Answer:
a) In the step-by-step explanation.
b) The null and alternative hypothesis are:
[tex]H_0: \mu=125000\\\\H_a:\mu> 125000[/tex]
c) P-value = 0.015
d) The conclusion, at a level of significance of 0.05, is that there is enough evidence to support the claim that that area residential property is available at a mean cost significantly higher than $125,000 per lot.
e) The claim of the chamber of commerce is false. We have statistical evidence against that claim and we can conclude that the mean prices are significantly higher than $125,000.
Step-by-step explanation:
This is a hypothesis test for the population mean.
The claim, expressed in the alternative hypothesis, is that area residential property is available at a mean cost significantly higher than $125,000 per lot.
Then, the null and alternative hypothesis are:
[tex]H_0: \mu=125000\\\\H_a:\mu> 125000[/tex]
The significance level is 0.05.
The sample has a size n=32.
The sample mean is M=130000.
As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=12500.
The estimated standard error of the mean is computed using the formula:
[tex]s_M=\dfrac{s}{\sqrt{n}}=\dfrac{12500}{\sqrt{32}}=2209.71[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{130000-125000}{2209.71}=\dfrac{5000}{2209.71}=2.26[/tex]
The degrees of freedom for this sample size are:
[tex]df=n-1=32-1=31[/tex]
This test is a right-tailed test, with 31 degrees of freedom and t=2.26, so the P-value for this test is calculated as (using a t-table):
[tex]\text{P-value}=P(t>2.26)=0.015[/tex]
As the P-value (0.015) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that that area residential property is available at a mean cost significantly higher than $125,000 per lot.
Using π = 3.14, find the area of a circle with a radius of 4.2. Round your answer to the nearest hundredth.
9514 1404 393
Answer:
55.39 square units
Step-by-step explanation:
The formula for the area of a circle is ...
A = πr²
Put the given numbers in place of the corresponding variables and do the arithmetic.
A = 3.14 × 4.2² = 3.14 × 17.64 = 55.3896
Rounded to hundredths, the area is ...
55.39 square units
5(- 3x - 2) - (x - 3) = -4(4x + 5) + 13 Solve
Answer:
Always true or 0 = 0
Step-by-step explanation:
This equation simplifies to 0 = 0 I will show in several steps.
1. (-15x - 10) - (x - 3) = (-16x - 20) + 13
2. -16x - 7 = -16x - 20 + 13
3. 0 = 7 - 20 + 13
4. 0 = 0
Hope this helps! (Please consider giving brainliest)
Answer:
all real value for x
Step-by-step explanation:
5(- 3x - 2) - (x - 3) = -4(4x + 5) + 13
Distribute
-15x -10 -x+3 = -16x -20 +13
Combine like terms
-16x-7 = -16x -7
Add 16x to each side
-16x-7 +16x= -16x -7+16x
-7 = -7
This is always true so x is all real values
Can you help me Please, hurry!!!!
Answer:
2,4,2,4.
Step-by-step explanation:
Answer:
2,4,2,4.
Step-by-step explanation:
Janet can wax her car in 2 hr(s). When she works together with Helga, they can wax the car in 45
min(s). How long would it take Helga, working by herself, to wax the car?
please help
Answer:
1.2h / 72min
Step-by-step explanation:
Maria waxes her car in 2 hours. When she works with Susan, they can wax the car
in 45 mins. How long will it take Susan, working by herself, to wax the car?
:
Let x = time required by Susan by herself
:
Change 45 min to hrs: 45/60 = .75 hrs
:
Let the completed job = 1
:
Each will do a fraction of the job and will add up to 1
:
.75%2F2 + .75%2Fx = 1
Multiply equation by 2x to eliminate the denominators, solve for x
.75x + 2(.75) = 2x
:
.75x + 1.5 = 2x
:
1.5 = 2x - .75x
:
1.5 = 1.25x
x = 1.5%2F1.25
x = 1.2 hrs (72min), for Susan to do the job by herself
A recent survey collected information on television viewing habits from a random sample of 1,000 people in the United States. Of those sampled, 37 percent indicated that their favorite sport to watch on television was American football. Construct and interpret a 95 percent confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football. Based on your answer to the previous problem, is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football
Answer:
The 95% confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football is (0.34, 0.40). This means that we are 95% sure that the true proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football is between (0.34 and 0.40).
The lower bound of the confidene interval is higher than 0.33. So it is not reasonable to believe that 33% is the actual percent of people in the United States whose favorite sport to watch on television is American football, this estimate should be higher than 33%.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
[tex]n = 1000, \pi = 0.37[/tex]
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.37 - 1.96\sqrt{\frac{0.37*0.63}{1000}} = 0.34[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.37 + 1.96\sqrt{\frac{0.37*0.63}{1000}} = 0.40[/tex]
The 95% confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football is (0.34, 0.40). This means that we are 95% sure that the true proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football is between (0.34 and 0.40).
Based on your answer to the previous problem, is it reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football
The lower bound of the confidene interval is higher than 0.33. So it is not reasonable to believe that 33% is the actual percent of people in the United States whose favorite sport to watch on television is American football, this estimate should be higher than 33%.
The 95% Confidence Interval we will found for given case is:
(0.34, 0.40)
It is Not reasonable to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football
Given that:Total size of sample = n = 1000Percent of people in sample indicating that their favorite sport on TV is to watch American football = p = 37% = 0.37 probability.To find:95% confidence interval for the proportion of all people in the United States who would indicate that their favorite sport to watch on television is American football.
Finding the Confidence Interval:When there is n sample size and the probability of success is p with level of significance [tex]\alpha[/tex]calculated as:
[tex]CI = p \pm z_0 \sqrt{\dfrac{p(1-p)}{n}[/tex]
Where [tex]z_0[/tex] is the z score [tex]1- \dfrac{\alpha}{2}[/tex] .
1 - [tex]\alpha[/tex] = 0.95 ( since it is given to find 95% CI)
Thus, [tex]\alpha[/tex] = 0.05
The z score at p value = 1 - [tex]\alpha[/tex]/2 = 1 -0.025 = 0.975 is 1.96
Thus we have:
[tex]CI = p \pm z_0 \sqrt{\dfrac{p(1-p)}{n}}\\\\CI = 0.37 \pm 1.96 \sqrt{\dfrac{0.37 \times0.63 }{1000}}\\\\CI = 0.37 \pm 1.96 \times 0.0152 = 0.37 \pm 0.03\\[/tex]
Lower limit of CI = 0.34 and Upper limit of CI = 0.40.
The case to decide to believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football can be solved as:
33% = 0.33 probability of actually liking watching american probability.
Since 0.33 is lower than the lower limit of CI for 0.37 probability, thus we reject the null hypothesis that both 0.37 and 0.33 are approximately same, and thus decide that it is false that 0.33 percent believe that 33 percent is the actual percent of people in the United States whose favorite sport to watch on television is American football.
Thus it is not reasonable to believe that 33% is the actual percent of people in the United States whose favorite sport to watch on television is American football.
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Parallel lines m and n are cut by transversal l. On line m where it intersects with line l, 4 angles are created. Labeled clockwise, from the uppercase left, the angles are: blank, blank, 45 degrees, blank. On line n where it intersects with line l, 4 angles are created. Labeled clockwise, from the uppercase left, the angles are blank, (2 x minus 5) degrees, blank, blank.
What is the value of x?
Answer:
X=70 :) Good Luck!!
Step-by-step explanation:
Using the concept of supplementary angles, it is found that the value of x is of 70 degrees.
-----------------
The situation is modeled by the graph at the end.The angles of 45º and (2x - 5)º are on the same side, thus they are supplementary.If two angles are supplementary, the sum of their measures is of 180º.Thus:
[tex]45 + 2x - 5 = 180[/tex]
[tex]2x + 40 = 180[/tex]
[tex]2x = 140[/tex]
[tex]x = \frac{140}{2}[/tex]
[tex]x = 70[/tex]
The value of x is of 70 degrees.
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Tina must select and answer in any order three of seven
seven essay questions on a test.
In how many ways can she do so?
Tina can select and answer the essay questions on her test in_____ ways.
(Simplify your answer. Type an integer or a fraction.)
Answer:
Tina can select and answer the essay questions on her test in 35 ways.
Step-by-step explanation:
The order in which she chooses the questions is not important. For example, choosing questions 1, 3 and 7 is the same as choosing 7, 3 and 1. So we use the combinations formula to solve this question.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
Three questions from a set of 7. So
[tex]C_{7,3} = \frac{7!}{3!(7-4)!} = 35[/tex]
Tina can select and answer the essay questions on her test in 35 ways.
Tina can select and answer the essay question on her test in 35 ways.
Number of ways to select 'r' objects out of 'n' objects is given by the expression,
Number of ways = [tex]^nC_r[/tex]
Since, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
If n = 7 and r = 3,
Number of ways to select 3 essays out of 7 will be,
Number of ways = [tex]^7C_3[/tex]
[tex]=\frac{7!}{3!(7-3)!}[/tex]
[tex]=\frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}[/tex]
[tex]=35[/tex]
Therefore, Tina can select and answer the essay question on her test in 35 ways.
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According to data from a medical association, the rate of change in the number of hospital outpatient visits, in millions, in a certain country each year from 1980 to the present can be approximated by f’(t) = 0.001155t(t-1980)^0.50, where t is the year.
a. Using the fact that in 1980 there were 264,034,000 outpatient visits, find a formula giving the approximate number of outpatient visits as a function of time.
b. Use the answer to part a to forecast the number of outpatient visits in the year 2015.
Answer:
a) [tex]f(t)=0.001155[\frac{2}{3}t(t-1980)^{3/2}-\frac{4}{15}(t-1980)^{5/2}]+264,034,000[/tex]
b) f(t=2015) = 264,034,317.7
Step-by-step explanation:
The rate of change in the number of hospital outpatient visits, in millions, is given by:
[tex]f'(t)=0.001155t(t-1980)^{0.5}[/tex]
a) To find the function f(t) you integrate f(t):
[tex]\int \frac{df(t)}{dt}dt=f(t)=\int [0.001155t(t-1980)^{0.5}]dt[/tex]
To solve the integral you use:
[tex]\int udv=uv-\int vdu\\\\u=t\\\\du=dt\\\\dv=(t-1980)^{1/2}dt\\\\v=\frac{2}{3}(t-1980)^{3/2}[/tex]
Next, you replace in the integral:
[tex]\int t(t-1980)^{1/2}=t(\frac{2}{3}(t-1980)^{3/2})- \frac{2}{3}\int(t-1980)^{3/2}dt\\\\= \frac{2}{3}t(t-1980)^{3/2}-\frac{4}{15}(t-1980)^{5/2}+C[/tex]
Then, the function f(t) is:
[tex]f(t)=0.001155[\frac{2}{3}t(t-1980)^{3/2}-\frac{4}{15}(t-1980)^{5/2}]+C'[/tex]
The value of C' is deduced by the information of the exercise. For t=0 there were 264,034,000 outpatient visits.
Hence C' = 264,034,000
The function is:
[tex]f(t)=0.001155[\frac{2}{3}t(t-1980)^{3/2}-\frac{4}{15}(t-1980)^{5/2}]+264,034,000[/tex]
b) For t = 2015 you have:
[tex]f(t=2015)=0.001155[\frac{2}{3}(2015)(2015-1980)^{1/2}-\frac{4}{15}(2015-1980)^{5/2}]+264,034,000\\\\f(t=2015)=264,034,317.7[/tex]