The spread of a virus is modeled by V (t) = −t 3 + t 2 + 12t,
where V (t) is the number of people (in hundreds) with the virus and t is the number of weeks since the first case was observed.
(a) Sketch V (t).
(b) What is a reasonable domain of t for this problem?
(c) Find the average rate of infection from t = 0 to t = 2.
(d) Find the instantaneous rate of infection as a function of t using the limit definition of the derivative.
(e) Find V (2) and V ‘ (2). Write a sentence interpreting V (2) and V ‘ (2) in terms of the number of infected people. Make sure to include units.
(f) Sketch the tangent line to the graph you drew in a. at the point (2, V (2)). State the slope of the tangent line.
(g) Use V (2) and V ‘ (2) to estimate the value of V (2.1).
(h) Find the maximum number of people infected at the same time and when the maximum occurs. Determine the rate of infection at this time.
Functions can be used to model real life scenarios
The reasonable domain is [tex]\mathbf{[0,\infty)}[/tex].The average rate of change from t = 0 to 2 is 20 persons per weekThe instantaneous rate of change is [tex]\mathbf{V'(t) = -3t^2 + 2t + 12}[/tex].The slope of the tangent line at point (2,V(20) is 10 The rate of infection at the maximum point is 8.79 people per weekThe function is given as:
[tex]\mathbf{V(t) = -t^3 + t^2 + 12t}[/tex]
(a) Sketch V(t)
See attachment for the graph of [tex]\mathbf{V(t) = -t^3 + t^2 + 12t}[/tex]
(b) The reasonable domain
t represents the number of weeks.
This means that: t cannot be negative.
So, the reasonable domain is: [tex]\mathbf{[0,\infty)}[/tex]
(c) Average rate of change from t = 0 to 2
This is calculated as:
[tex]\mathbf{m = \frac{V(a) - V(b)}{a - b}}[/tex]
So, we have:
[tex]\mathbf{m = \frac{V(2) - V(0)}{2 - 0}}[/tex]
[tex]\mathbf{m = \frac{V(2) - V(0)}{2}}[/tex]
Calculate V(2) and V(0)
[tex]\mathbf{V(2) = (-2)^3 + (2)^2 + 12 \times 2 = 20}[/tex]
[tex]\mathbf{V(0) = (0)^3 + (0)^2 + 12 \times 0 = 0}[/tex]
So, we have:
[tex]\mathbf{m = \frac{20 - 0}{2}}[/tex]
[tex]\mathbf{m = \frac{20}{2}}[/tex]
[tex]\mathbf{m = 10}[/tex]
Hence, the average rate of change from t = 0 to 2 is 20
(d) The instantaneous rate of change using limits
[tex]\mathbf{V(t) = -t^3 + t^2 + 12t}[/tex]
The instantaneous rate of change is calculated as:
[tex]\mathbf{V'(t) = \lim_{h \to \infty} \frac{V(t + h) - V(t)}{h}}[/tex]
So, we have:
[tex]\mathbf{V(t + h) = (-(t + h))^3 + (t + h)^2 + 12(t + h)}[/tex]
[tex]\mathbf{V(t + h) = (-t - h)^3 + (t + h)^2 + 12(t + h)}[/tex]
Expand
[tex]\mathbf{V(t + h) = (-t)^3 +3(-t)^2(-h) +3(-t)(-h)^2 + (-h)^3 + t^2 + 2th+ h^2 + 12t + 12h}[/tex][tex]\mathbf{V(t + h) = -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h}[/tex]
Subtract V(t) from both sides
[tex]\mathbf{V(t + h) - V(t)= -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h - V(t)}[/tex]
Substitute [tex]\mathbf{V(t) = -t^3 + t^2 + 12t}[/tex]
[tex]\mathbf{V(t + h) - V(t)= -t^3 -3t^2h -3th^2 - h^3 + t^2 + 2th+ h^2 + 12t + 12h +t^3 - t^2 - 12t}[/tex]
Cancel out common terms
[tex]\mathbf{V(t + h) - V(t)= -3t^2h -3th^2 - h^3 + 2th+ h^2 + 12h}[/tex]
[tex]\mathbf{V'(t) = \lim_{h \to \infty} \frac{V(t + h) - V(t)}{h}}[/tex] becomes
[tex]\mathbf{V'(t) = \lim_{h \to \infty} \frac{ -3t^2h -3th^2 - h^3 + 2th+ h^2 + 12h}{h}}[/tex]
[tex]\mathbf{V'(t) = \lim_{h \to \infty} -3t^2 -3th - h^2 + 2t+ h + 12}[/tex]
Limit h to 0
[tex]\mathbf{V'(t) = -3t^2 -3t\times 0 - 0^2 + 2t+ 0 + 12}[/tex]
[tex]\mathbf{V'(t) = -3t^2 + 2t + 12}[/tex]
(e) V(2) and V'(2)
Substitute 2 for t in V(t) and V'(t)
So, we have:
[tex]\mathbf{V(2) = (-2)^3 + (2)^2 + 12 \times 2 = 20}[/tex]
[tex]\mathbf{V'(2) = -3 \times 2^2 + 2 \times 2 + 12 = 4}[/tex]
Interpretation
V(2) means that, 20 people were infected after 2 weeks of the virus spread
V'(2) means that, the rate of infection of the virus after 2 weeks is 4 people per week
(f) Sketch the tangent line at (2,V(2))
See attachment for the tangent line
The slope of this line is:
[tex]\mathbf{m = \frac{V(2)}{2}}[/tex]
[tex]\mathbf{m = \frac{20}{2}}[/tex]
[tex]\mathbf{m = 10}[/tex]
The slope of the tangent line is 10
(g) Estimate V(2.1)
The value of 2.1 is
[tex]\mathbf{V(2.1) = (-2.1)^3 + (2.1)^2 + 12 \times 2.1}[/tex]
[tex]\mathbf{V(2.1) = 20.35}[/tex]
(h) The maximum number of people infected at the same time
Using the graph, the maximum point on the graph is:
[tex]\mathbf{(t,V(t) = (2.361,20.745)}[/tex]
This means that:
The maximum number of people infected at the same time is approximately 21.
The rate of infection at this point is:
[tex]\mathbf{m = \frac{V(t)}{t}}[/tex]
[tex]\mathbf{m = \frac{20.745}{2.361}}[/tex]
[tex]\mathbf{m = 8.79}[/tex]
The rate of infection is 8.79 people per week
Read more about graphs and functions at:
https://brainly.com/question/18806107
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Tìm đạo hàm Y' = Y'(x) tại X0 = π/4 của hàm số y -= y(x) được cho bởi phương trình tham số
X= arctan t
y = ln t
about it ok to my house and Elsa and Anna and I are going to my house and I don't know what it's like to my
a square has a perimeter of 6x - 20 units. What is the value of x?
The amount of time it takes a swimmer to swim a race varies inversely as the average speed of the swimmer. A swimmer finishes a race in 37.5 seconds with an average speed of 4 feet per second. Find the average speed of the swimmer if it takes 25 seconds to finish the race.
Answer:
5 average speed
Step-by-step explanation:
Answer:
eeessddsddddddddd
Step-by-step explanation:
ddds
what is the answer to
5x=40
Answer:
X has to be 8, so 5x8=40
Step-by-step explanation:
can someone plz help me with this
Answer:
option D .....11
Option c 36
Pls somone tell
Me the answer plzzzz
Answer:
POSITIVE
Step-by-step explanation:
Because the product of 2 negative numbers is a positive number
Last quarter Rachel earned a grade of 92. This quarter she earned an 88. What is the percent decrease in her grade?
Answer:
[tex]$4.\overline{54}[/tex]
Step-by-step explanation:
The percent decrease is [tex]$92/88-1=23/22-1=1/22=4.5454545...$[/tex].
So, the answer is [tex]$\boxed{4.\overline{54}}$[/tex] percent decrease.
On five days of every week Jackie runs 2 1/2 miles in the morning how many total miles is Jackie run every week
Answer:
so I believe the question is just wanting you to do 5 times 2 1/2. I believe the answer would be 12 1/2. Your just adding 2 1/2, 5 times for each day she runs.
what is the average of 2 1/3 and 3 1/2
Answer:
2 11/12
Step-by-step explanation:
The average of 2 ⅓ and 3 ½ is 2 11⁄12. To find the average, find the sum of the fractions and divide by 2.
2 ⅓ + 3 ½ = 7⁄3 + 7⁄2 = 14⁄6 + 21⁄6 = 35⁄6 or 5 5⁄6
5 5⁄6 ÷ 2 = 35⁄6 × 1⁄2 = 35⁄12 or 2 11⁄12
The average of 2⅓ and 3½ is 2 11/12
2⅓ + 3½ = 7/3 + 7/2 = 14/6 + 21/6 = 35/6 or 5/6
5 5/6 ÷ 2 = 35/6 × 1/2 = 35/12 or 2 11/12
Finding the Average
Average refers to the sum of the values divided by the count. Therefore, the steps to follow to find the average are below:
Add the given values.
Divide the sum by the count of values.
Equation:
(2 1/3 + 3 1/2) / 2 = ?
Solution:
(2 1/3 + 3 1/2) / 2 = ?
* Change the mixed numbers to improper fractions.
= (7/3 + 7/2) / 2
* Add the fractions. Find the LCD.
= (14/6 + 21/6) / 2
= ( 35/6 ) /2
To divide change the sign to multiplication, The give the reciprocal of the second fraction
= 35/6 × 1/2
= 35/12 or 2 11/12
Final Answer:2 11/12
hope it helps youThank You !!five times a number is 120, what is the number
Answer:
Step-by-step explanation:
24
If 2.5 kg of potatoes cost $8.25 how much will u pay for 7 kg
Answer:
$23.10
Step-by-step explanation:
To get how much money you will have to pay for 7kg, you need to find out how much you have to pay for 1 kg. You will divided y divided by x = y.
8.25 (y) divided by 2.5 (x) = $3.30
You will pay $3.30 for 1 kg
So $3.30 x 7 = 23.10
You will pay $23.10 for 7 kg of potatoes.
** I hope this helps
Helen wants to paint a rectangle with an area of
35 square feet on her wall. If the height of the rectangle is 3 feet,
what is the width of the rectangle? (Remember, area = length x width.)
Write an equation and solve.
hannah and mallory are selling boxes of cookies for a school fundraiser. each box costs the same amount. hannah sells 14 boxes. mallory sells 22 boxes. mallory's neighbor makes a $6 donation without buying any cookies. together, hannah and mallory raise $141. write ans solve an equation to find out how much each box of cookies costs. how much does hannah raise? how much does mallory raise?
The cost per box of cookies is $3.75
The total amount Hannah raised from the sales of cookies is $52.50
The total amount Mallory raised from the sales of cookies and donations is $88.50
In order to determine the cost per box of cookies, take the following steps:
The first step is to determine the total sum of the boxes sold by Hannah and Mallory
Sum of the boxes sold = 14 + 22 = 36 boxes
36 boxes were sold by Hannah and Mallory
The second step is to determine the total cost of the 36 boxes sold.
Total cost of 36 boxes = total amount raised - donation
$141 - $6 = $135
The third step is to determine the total cost of each box
cost of one box = total cost of 36 boxes / total number of boxes
135 / 36 = $3.75
The total amount Hannah raised = cost per box x number of boxes she sold
14 x $3.75 = $52.50
The total amount Mallory raised = amount raised from the sale of cookies + donation
amount raised from sales = cost per box x number of boxes she sold
22 x $3.75 = $82.50
Total amount = $82.50 + $6 = $88.50
A similar question was answered here: https://brainly.com/question/10915476?referrer=searchResults
Write a quadratic equation that has 2 and 8 as solutions.
Find the equation with 1 as the coefficient of the quadratic term.
Answer:
[tex]x^2-10x+16[/tex]
Step-by-step explanation:
The way I would find the equation is by finding the factored form of the equation form. That equation would be (x-2)(x-8). These are correct because if you set both of these equal to zero, your x-values will be equal to 2 and 8.
x-2=0
x=2
x-8=0
x=8
If you want the standard form of the equation, all you have to do is multiply it out.
[tex](x-2)(x-8)\\x(x-8)-2(x-8)\\x^2-8x-2x+16\\x^2-10x+16[/tex]
Boom. There's your answer!
5339 x 6 and explain
32,034
Step-by-step explanation:
5,339
6 ×
32,034
9 × 6 = 54 write 4, remainder 53 × 6 = 18 plus 5 = 23 write 3, remainder 23 × 6 = 18 plus 2 = 20 write 0, remainder 25 × 6 = 30 plus 2 = 32i will give brainliest to the right answers
please help
Answer:
first answer = 2
second answer = 12
third answer = - 12
fourth answer = 2
1/5(x + 0.5) + 5.24 = 3/2x + 7/10(x + 2.2)
Solve for x
Answer:
x=19/10
Step-by-step explanation:
abcdefghijklmnopqrstuvwxyz
A recipe calls for 9 1/2 cups of flour to make a batch of cookies. How many batches of cookies can be made if you have 38 cups of flour?
Who can answer this?
Answer:
because of parallel lines it's 63
If p<0,q>0 and q=r what statement about is r true
Answer:
hell no
Step-by-step explanation:
bbbb
hello there looking for help here thanks. giving brainliest if able to
Answer:
angles 2 and 10 are corresponding Angles while angles 6 and 11 are alternate Angles
Answer:
1. False Same side exterior angles refers to same side transversal An example; opposite ends both same side of transversal. see 2nd attachment.
2. False interior alternate are both interior alternate on a Z 3 would be alternate with 6 as both are within the z and diagonally opposite at ends of the z shape.
3. True (yes both same sides of transversal one is left of parallel line the other is left of the next parallel.
4. False ( 12 is only alternate with 1 as alternate is within the z shape as exterior and opposite sides of the transversal)
Step-by-step explanation:
Alternate Interior Angles – Alternate interior angles are the pair of angles on the inner side of the two parallel lines but on the opposite side of the transversal. Alternate Exterior Angles – Alternate exterior angles are the pair of angles on the outer side of the two parallel lines but on the opposite side of the transversal
Same side exterior angles - are seen on the blue spots on the 2nd diagram transversal
Solve the photo above ⬆️
Btw these one's are gonna be a bit more confusing for me, so expect me to atleast probably send 7 more ^^|l|l
Answer:
7¹⁰ :) hope this helps you!!
In ΔJKL, j= 2.5 cm, k = 9.3 cm and l =8.4 cm. Find the area of ΔJKL to the nearest square centimeter
Answer:10
Step-by-step explanation:
A triangle has side lengths of 14 cm and 7 cm. Which could be the value of the third side, 22 cm or 15 cm?
Answer: 15 cm
Step-by-step explanation: So first, do 14 squared plus 7 sqaured. The answer is 245. And now let’s compare it to 22 cm and 15 cm. 22 squared is 484 and 15 sqaured is 225. 225 is closer to 245 than 484. Therefore, 15 is the correct answer.
help me make a apple account and put the user and password please
what would the coordinates (10,-3) lie on?
HELP. WILL MAKE BRAINLIEST Make a true congruence statement about the following triangles:
linked in picture
How will I answer that, the image isn't showing luv
PLEASE EXPLAIN AND PLEASE HELP ME AND EXPLAIN I NEED HELP
its right because
6 x 5 is the area of the leftmost rectangle
4 x 5 is the area of the bottom rectangle
3 x 5 is the area of the rightmost rectangle
5 x 5 is the area of the slanted square on the top
the last term is the area of the trapezoid, which is (base1 + base2) x height /2
they split the trapezoid thing into a square and a triangle, which gets you the same answer
they multiplied it by 2 because there are 2 trapezoids
