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.
Answers
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
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Related Questions
please answer this, i am so confused
Answers
The bag of dog food would last for 21 days.
The bag contains 42 cups of dog food. If the dog eats [tex]2\frac{1}{3}[/tex] cups everyday, the amount of cup in the bag decreases by [tex]2\frac{1}{3}[/tex] cups everyday. This can be represented with this expression: 42 - [tex]2\frac{1}{3}[/tex]d. Where d represents the number of days.
In order to determine the number of days in which the bag would be emptied, this equation can be used:
Number of days the dog food would last = number of cups in the cup / amount the dog eats each day
42 ÷ [tex]2\frac{1}{3}[/tex]
42 ÷ [tex]\frac{7}{3}[/tex]
42 x [tex]\frac{3}{7}[/tex] = 21 days
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pls help asap i rly need this
Answers
Answer:
THey are countin by 2 and 5 on the graph
Step-by-step explanation:
SOMEONE HELP ME AND EXPLAIN HOW ITS THE ANSWER PLZ!!!
Answers
Answer:
20 guests
Step-by-step explanation:
20 (20) + 400 = 400 + 400 = 800
10 (20) + 600 = 200 + 600 = 800
answer B = 20 guests
Answer:
For 20 persons the price would be same.
Step-by-step explanation:
y=y
20x+400=10x+600
20x-10x=600-400
10x=200
x=20
Therefore, for 20 persons the price would be same.
Which expression is equivalent to 1/5(150x-80y+50x-25y+20)
Answers
Answer: 40x-21y+4
Step-by-step explanation:
[tex]\frac{1}{5}(150x-80y+50x-25y+20)\\\\=\frac{1}{5}(200x-105y+20)\\\\ =\boxed{40x-21y+4}[/tex]
is the product of 2 numbers always the LCM
Answers
Answer
I guess so.
To find the LCM you need to know the lowest digit as possible.
For example,
Question: What is the LCM of 8 and 6?
Answer: Table of 8: 8, 16, 24, 32, 40,48
Table of 6: 6, 12, 18, 24, 30, 36, 42, 48
Now look at the tables again and, did you see the LCM???......
If so your answer would be 24.
Step-by-step explanation:
Select the correct answer.
Sam is moving bags of soil from a loading dock into a storeroom. The topsoil weighs 40 pounds per bag, and the mulch weighs 20 pounds per
bag. The cart Sam uses to move the bags can only carry up to 480 pounds.
Which graph represents the number of topsoil bags and mulch bags that Sam can move in a single trip?
Answers
Answer:
The answer would be the A (I included a picture to show you which one it is)
Step-by-step explanation:
Options D and E are out because it clearly says " bags can only carry up to 480 pound" and those are saying it can hold more than 480
Options B and C have a dashed line. If it has a dashed line that means that it is < or > and it is not because it says "up to" up to means it CAN hold exactly 480
So the only right answer would be Option A
Answer:
A
Step-by-step explanation:
Got it right off Edmentum!!
On a road map the distance from Cily A to City BIS 4 inches. The map
scale is 1 inch 20 miles what is the actual distance in miles, between
City A and City B2
5 miles
24 miles
8 miles
80
Answers
Answer:
80 miles
Step-by-step explanation:
The map scale compares the distance between two points on the map with the corresponding distance on the ground.
That is,
[tex]\boxed{\mathfrak{map\:scale=\frac{distance\:on\:map}{distance\:on\:ground} }}[/tex]
The formula makes it pretty easy to solve!
map scale is 1 inch : 20 milesdistance on map is 4 inchesLet's say, the corresponding distance on the ground is x miles.Substituting these values in the above mentioned formula:
[tex]\implies \mathsf{\frac{1 \:inch}{20\:miles} =\frac{4\:inches}{x\:miles} }[/tex]
cross multiplying:
[tex]\implies\mathsf{x=4 \times 20}[/tex]
[tex]\implies \mathsf{x=80}[/tex]
That makes the actual distance 80 miles!
ALTER
You can also use simple unitary method here:
1 inch on the map = 20 miles on the ground
∴ 4 inches on the map = 4 × 20 miles on the ground
= 80 miles on the ground
Which equation represents the line shown in the graph below?
у
+
8
6
4
2
X
-8
-6
-21
6
2
8
2
-6
-8
O A. y = 3x + 4
OB.y = 31 - 4
OC. y = 4x + 3
OD
y = 4.1 - 3
Answers
The equation of line from the graph is y = 3x + 4
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
The equation y = 3x + 4 represents a line in the coordinate plane.
To graph this line, we can plot a few points
when x=0, y=3(0)+4=4,
So we have the point (0,4)
when x=1, y=3(1)+4=7,
So we have the point (1,7)
when x=-2, y=3(-2)+4=-2,
So we have the point (-2,-2)
Plotting these points and connecting them with a straight line, we get the graph of the line
Hence , the equation of line is y = 3x + 4
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A = 1/2 h(b^1 + b^2) solve for b^2
Answers
Answer:
[tex]b_{2} = \frac{2A}{h} - b_{1}\\[/tex]
Step-by-step explanation:
[tex]A = \frac{1}{2} h(b_{1} +b_{2})\\2A = h(b_{1} +b_{2})\\\frac{2A}{h} = b_{1} +b_{2}\\\frac{2A}{h} - b_{1} = b_{2}\\[/tex]
he owner of a coffee shop collected receipts from all orders placed during a one-hour period. For each receipt, the owner recorded the payment method used and if the receipt was $25 or less or more than $25. The following table shows the requencies of the results.
Payment
Cash Credit/Debit Phone App Total Receipt of $25 or less 15 25 17
Receipt of more than $25 23 37 28
Total 145
What is the probability that a receipt that is to be selected at random from those collected during the one-hour period was
paid using a phone app, given that the receipt totaled more than $25?
Enter your answer as a value between 0 and 1 rounded to two decimal places. Points A,
Answers
Answer:
56
Step-by-step explanation:
The probability of selecting a receipt of more than $25 is 0.33.
What is a probability?Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.
For the given situation,
The amount on receipts in $, S= { 15,25,17,23,37,28 }
⇒ [tex]n(s) = 6[/tex]
The amounts are grouped as more than $25 and less than $25.
The event is the amount on receipt more than $25, [tex]e = (28,37)[/tex]
⇒ [tex]n(e)=2[/tex]
The formula to find the probability of an event,
[tex]P(e)=\frac{n(e)}{n(s)}[/tex]
⇒ [tex]P(e)= \frac{2}{6}[/tex]
⇒ [tex]P(e)=\frac{1}{3}[/tex]
⇒ [tex]P(e)=0.33[/tex]
Hence we can conclude that the probability of selecting the receipt of more than $25 is 0.33.
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Read The Question in the PNG
Answers
9514 1404 393
Answer:
A. dollars = euros × 0.75
B. the exchange rate is the slope of the graph
Step-by-step explanation:
A. The graph indicates 1 euro will be exchanged for 0.75 dollars, or 4 euros for 3 dollars.
__
B. This exchange rate is the slope of the line on the graph: 0.75 dollars per euro.
a) 16 / (-4)
b) (-56) / 8
c) (-28) / (-2)
Answers
Or is that the question?
f(x)=3x 3 inverse function
Answers
Step-by-step explanation:
if I understand this correctly than you are looking for the inverse function of f(x) = y = 3x³.
the inverse function simply tries to calculate the original x it of the original y.
and then, to make it a formal function, we rename x to y and y to x.
y = 3x³
x³ = y/3
[tex]x = \sqrt[3]{ \frac{y}{3} } [/tex]
=> as "regular" function
[tex]y = {f}^{ - 1} (x)= \sqrt[3]{ \frac{x}{3} } [/tex]
57²⁰¹⁸ and 51²⁰²⁰, which one is larger? (57^2018 vs 51^2020)
Answers
Answer:
steps below
Step-by-step explanation:
Set log x = a and log y = b
x = 10ᵃ and y = 10ᵇ
if a>b 10ᵃ > 10ᵇ
so x > y
x = 57²⁰¹⁸ y = 51²⁰²⁰
log x = log 57²⁰¹⁸ = 2018 * log 57 = 3543.36 : (a)
log y = log 51²⁰²⁰ = 2020 * log 51 = 3449.29 : (b)
3543.36 > 3449.29 log x > log y
x > y: 57²⁰¹⁸ > 51²⁰²⁰
Express the distance 4,300,000 meters using scientific notation in meters, and then in millimeters.
Answers
Answer: 4.3 × 10^6 meters, and 4.3 × 10^9 millimeters.
Explanation: In order to find the scientific notation of a large number, you want to move the decimal point to the left until you only have one digit to the left of the number, then you can count the spaces you have moved, and this will make the power of 10 that you'll add to the end. This seems a little confusing, but let's look at it through the example.
Step 1) Move the decimal point to the left until there's only one digit to the left of the decimal:
4,300,000 turns into 4.300000
Step 2) Count the number of digits you've moved.
Above, we moved 6 spaces to the left.
Step 3) Add the decimal to the front, and then 10^the number of spaces you moved
4.3 (from step one) × 10^6 (from step two)
= 4.3 × 10^6
It works the same way for the millimeter question, just convert meters into millimeters before you start the problem, and then solve it using the steps above! I hope this helped you :)
The scientific notation in meters and millimeters are
- 4.3 x [tex]10^{6}[/tex] meters
- 4.3 x [tex]10^{9}[/tex] millimeters
What is scientific notation?It is a way of writing very large or very small numbers where a number between 1 and 10 is multiplied by a power of 10.
Example:
310,000,000 can be written in scientific notation as 3.1 ✕ 10^8.
We have,
Distance = 4,300,000 meters
1 meter = 1000 millimeters
So,
4,300,000 meters = 4,300,000,000 millimeters
We can write 4,300,000 in scientific notation as:
= 4.3 x [tex]10^{6}[/tex] meters
We can write 4,300,000,000 millimeters in scientific notation as:
= 4.3 x [tex]10^{9}[/tex] millimeters
Thus the scientific notation in meters and millimeters are
- 4.3 x [tex]10^{6}[/tex] meters
- 4.3 x [tex]10^{9}[/tex] millimeters
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PLEASE HELP ITS MATH THANK YOUUUU
Answers
Answer:
6 ft
Step-by-step explanation:
Multiply by 2 on each side: (8*2) = (3*2)
New ratio: 16 = 6
Final answer: 6ft
Two side lengths of a triangle are 30.9 inches and 42.0 inches. the sides meet at a 19degree angle. What is the area of the triangle?
Answers
Step-by-step explanation:
A=hbb
=42· 0 x30.9
=648.9
Simplify, the solve.
Answers
Answer:
1/64
Step-by-step explanation:
4x4^3/4^7
4^-3
1/4^3
1/64
On a map, 1 inch equals 25 miles if two cities are 8 inches apart on the map how far are they actually apart ?
Answers
Answer:
200 mi
Step-by-step explanation:
8 times 25 mi = 200 mi
PLEEEEEEEEEAAAse mark branliest!!!
Multiply. (−6/35)⋅14/27 What is the product? Enter your answer as a fraction, in simplified form, in the box.
Answers
Answer:
-4/45
Step-by-step explanation:
[tex]\dfrac{-6}{35}*\dfrac{14}{27}=\dfrac{-6}{5}*\dfrac{2}{27}[/tex]
14 and 35 are given by 7th table.
[tex]= \dfrac{-2}{5}*\dfrac{2}{9}[/tex]
Now , 6 and 27 are given by 3rd table
[tex]=\dfrac{-4}{45}[/tex]
Answer:
-4/45
Step-by-step explanation:
2.03 Quiz: Multiply Signed Fractions
Parallelogram JKLM is reflected across a horizontal line through its center and then translated down and to the left to produce parallelogram QTSR.
Answers
Answer:
jjztztjsyiydkkyddyoydoxhkyxyxi
Step-by-step explanation:
hmxkhxkyxhmxykxm
PLEASE HELP ME ITS ALGEBRA THANK YOUUUUUUU
Answers
please help me solve this !
will mark brainliest !
(spammers reported)
Answers
Answer:
it is five
Step-by-step explanation:
5v-2=8 then v=x2 because 5x2=10-2=8
so 7v-9=5
The correct answer is 5
Jaliyah wants to play a prank on her older sister at her birthday party. She has two bags of blue glitter that she's planning on filling a birthday hat with. To figure out how much glitter she needs to use, Jaliyah must first find the volume of the party hat. Find the volume of the party hat in cubic inches.
Height = 8.4 inches
Length = 5.3 inches
Answers
Answer:
44.52
Step-by-step explanation:
Volume = b * h
base times height
The height is 8.4 inches so 8.4 *
Length is the same as base so 5.4
8.4 * 5.4
This table shows the relationship between time and push-ups.
Which of the following values could represent the value of r2 for this relationship?
r2 = ‒0.9646
r2 = ‒0.0354
r2 = 0.0354
r2 = 0.9646
Answers
Answer:
D
Step-by-step explanation: i <3 satistics so much
plss helpppppppppppp plssss
Answers
Answer:
(x + 1)(x - 7) is the odd one.
Step-by-step explanation:
1) sum= 8
Product = 7
Factors = 1 , 7 {1*7 = 7 & 1+7 = 8}
x² + 8x + 7 = x² + x + 7x + 7
= x(x + 1) + 7(x + 1)
=(x +1 )(x + 7)
2) x² - 8x + 7 = x² -x -7x + (-1)*(-7)
= x(x - 1) - 7(x - 1)
=(x - 1)( x - 7)
3)Sum = 6
Product = -7
Factors = (-1) , 7 { -1 + 7 = 6 & (-1)*7 = -7}
x² + 6x - 7 = x² - x + 7x - 7
= x(x - 1) + 7(x - 1)
= (x - 1)(x + 7)
Answer:
(x+1)(x-7) is the odd one out
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Answers
answer: the inequality is 3+7(x)<24
x=x < 3 2/7
Step-by-step explanation:
because...
3 + 7x < 26
7x < 23
x < 23/7
x < 3 2/7
Answer:
More/Product is a key word to multiply so 3X7 and less is a key word to subtract so your subtracting X from 24
3x7=21 and that's 3 less than 24
The answer is 3
Step-by-step explanation:
A giant thresher shark weighing 1250 pounds, believed to be the heaviest ever caught, was landed by fisherman Roger Nowell off the Cornish coast in November 2007. The fish was sold by auction at Newlyn Fish Market for £255. Roughly, what was the cost per pound?
Answers
Which expression has a solution of 54, if t = 6? 7 t 9 t 6 t 8 t
Answers
Answer:
The answer is 9
Step-by-step explanation:
Cause when we multiplied 6 to 9 the answer will be 54
Calculus, derivatives. Please help! Show work, if possible. Thanks! :)
Answers
Answer:
a. y=4
b. graph the given and y=4
Step-by-step explanation:
The given is y= 8 sin(x) cos(x)
to get y' which is the slope of the tangent line we take the derivative
for this problem, the derivative involves the constant rule (which states that the derivative a constant times a value is just the constant times the derivative of said value) and the multiplication rule (assuming that the two parts in question are represented by the variables a and b then the derivative of a*b = a*b'+b*a')
thus we get 8*(sin(x)*-sin(x)+cos(x)*cos(x))
-8sin^2(x)+8cos^2(x)
You can also leave the 8 differentiated out depending on the form you need.
Now that we have the derivative we can plug pi/4 into this equation to get the slope of the tangent line at that point
=8sin^2(pi/4)+8cos^2(pi/4)
this would equal -8*1/2 +8*1/2 which equals 0
the slope of the tangent at this point would be zero
To get the point for point slope form we plug the pi/4 into the given equation to get a y value and get the point (pi/4, 4)
this means the point slope equation would be
y-4=0(x-pi/4)
y-4=0
y=4
To graph you would just enter the given equation and the equation for the tangent line into a graphing software or calculator and see if the line is tangent at the point (pi/4, 4)