Verify that the indicated family of functions is a solution of the given differential equation. dP/dt = P(1-P); P = ce^t / 1+ ce^t?
Answers
The value for differential equation is found as -
[tex]\frac{dP}{dt} =\frac{d}{dt} (\frac{c_1e^t}{1+c_1e^t})\\[/tex]
[tex]\frac{dP}{dt} - (\frac{c_1e^t}{(1+c_1e^t)} )(1-\frac{c_1e^t}{(1+c_1e^t)^2})\\ =(\frac{c_1e^t}{(1+c_1e^t)^2} )-(\frac{c_1e^t}{1+c_1e^t})(1-\frac{c_1e^t}{1+c_1e^t})=0[/tex]
What is differential equation?
Any equation with one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation.
The equation given is - [tex]P=\frac{c_1e^t}{1+c_1e^t}[/tex]
Take derivative with respect to t -
[tex]\frac{dP}{dt} =\frac{d}{dt} (\frac{c_1e^t}{1+c_1e^t})\\\frac{dP}{dt} =\frac{d}{dt}c_1 (\frac{e^t}{1+c_1e^t})[/tex]
By Quotient rule [tex]\frac{d}{dt} \frac{u}{v} =\frac{v\frac{du}{dt}- u\frac{dv}{dt}}{v^2}[/tex] -
[tex]\frac{dP}{dt} =c_1 (\frac{(1+c_1e^t)\frac{d}{dt}(e^t)-(e^t)\frac{d}{dt}(1+c_1e^t)}{(1+c_1e^t)} )\\\frac{dP}{dt} =c_1 (\frac{(1+c_1e^t)(e^t)-(e^t)(0+c_1e^t)}{(1+c_1e^t)} )[/tex]
([tex]\frac{d}{dx} e^t=e^t[/tex] and [tex]\frac{d}{dx} c_1=0[/tex] here [tex]c_1=[/tex]constant)
[tex]\frac{dP}{dt} =c_1e^t (\frac{(1+c_1e^t-c_1e^t)}{(1+c_1e^t)^2} )\\\frac{dP}{dt} = (\frac{(c_1e^t)}{(1+c_1e^t)^2} )[/tex]
Now, [tex]\frac{dP}{dt} =P(1-P)[/tex] -
[tex]\frac{dP}{dt} - (\frac{c_1e^t}{(1+c_1e^t)} )(1-\frac{c_1e^t}{(1+c_1e^t)^2})\\ =(\frac{c_1e^t}{(1+c_1e^t)^2} )-(\frac{c_1e^t}{1+c_1e^t})(1-\frac{c_1e^t}{1+c_1e^t})\\\frac{dP}{dt} =(\frac{c_1e^t}{(1+c_1e^t)} )(\frac{1}{1+c_1e^t}-1+\frac{c_1e^t}{1+c_1e^t})\\\frac{dP}{dt} =(\frac{c_1e^t}{(1+c_1e^t)} )(\frac{1-(1+c_1e^t)+c_1e^t}{1+c_1e^t})\\\frac{dP}{dt} =(\frac{c_1e^t}{(1+c_1e^t)} )(\frac{1-1-c_1e^t+c_1e^t}{1+c_1e^t})\\[/tex]
[tex]\frac{dP}{dt} =(\frac{c_1e^t}{(1+c_1e^t)} )(\frac{0}{1+c_1e^t})\\\frac{dP}{dt} =(\frac{c_1e^t}{(1+c_1e^t)} )(0)\\\frac{dP}{dt} =0[/tex]
Therefore, the value is [tex]\frac{dP}{dt} =0[/tex].
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Related Questions
Are the two triangles similar? If so, state the reason and the similarity statement.
A) Impossible to determine
B) The triangles are not similar.
C) Yes, SSS;VWX ~ YZX
D)Yes, SAS; VWX~YZX
Answers
The ratio of the corresponding sides are different. Then the triangles are not similar. Then the correct option is B.
What is the triangle?The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.
The ratio of the matching sides will remain constant if two triangles are comparable to one another.
The ratio of the corresponding sides are given as,
YX / XV = ZX / XW
3 / 9 = 2 / 8
1 / 3 ≠ 1 / 4
The ratio of the corresponding sides are different. Then the triangles are not similar. Then the correct option is B.
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Select all that apply.
Given the points (-6, 8) and (-3, 4), which of the following are true about the line passing through these points?
The line represents a direct variation function.
The line has a slope of - 3/4.
The point (9, -12) is also on the line.
The line has a slope of 3/4 .
Answers
Answer:
The line has a slope of - 3/4.
The line has a slope of 3/4.
Step-by-step explanation:
To determine the slope of a line passing through two points, you can use the formula:
Slope = (y2 - y1) / (x2 - x1)
Using the points (-6, 8) and (-3, 4), the slope would be:
Slope = (4 - 8) / (-3 - (-6)) = -4 / 3 = -4/3
So, the line has a slope of -4/3. The other statements in the list are not necessarily true based on the given information.
1) Find the perimeter of the composite figure.
A) 23.5
B) 42.9
C) 46.9
D) 31.9
2) Find the area of the composite figure.
A) 64.25
B) 42.95
C) 39.5
D) 103.5
Answers
1) The perimeter of the composite figure is 31.9
2) The area of the composite figure is 64.25
What is perimeter?
The complete length of a shape's edge serves as its perimeter in geometric terms. Adding the lengths of all the sides and edges that surround a form yields its perimeter. It is calculated using linear length units such centimeters, meters, inches, and feet.
There are 2 shapes: half circle and right triangle
diameter of half circle ,d=10
radius, r=d/2 = 10/2 = 5
Perimeter of half circle = π r = 3.14 * 5 = 15.7
In the right triangle,
base = 5, line perpendicular to base = 10
hypotenuse² = squares of other 2 sides added
= 5² + 10²
= 25 +100
= 125
hypotenuse = ± √125 = ±11.18
consider positive for length
hypotenuse = 11.18
The perimeter of the composite figure
= Perimeter of half circle +base of right triangle + hypotenuse of right triangle
=15.7 + 5 + 11.18
= 31.9
Area of half circle = 0.5*π *r² = 0.5*3.14*5² = 39.25
Area of right triangle = 0.5*base * height
= 0.5*5*10
= 25
The area of the composite figure
=Area of half circle + Area of right triangle
= 39.25 + 25
= 64.25
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The monthly dues for a premium membership at a health club is $15 more than the cost of a standard membership.
The premium membership is $40 per month.
What is the cost of a standard membership?
please explain how you got the answer
Answers
Answer:
To find the cost of a standard membership, we need to subtract the $15 premium from the monthly premium membership cost of $40.
$40 - $15 = $25
Therefore, the cost of a standard membership is $25 per month.
Diane asked 12 students how many courses they have taken so far at her college. Here is the list of answers. 14, 3, 22, 10, 3, 5, 23, 18, 16, 9, 1, 26 What is the percentage of these students who have taken fewer than 4 courses?
Answers
There are 3 numbers listed that are less than 4 so 3/12= .25 and move two places to the right or x100 and get 25%
why does m1 x m2 = -1
Answers
Answer:
By Thales' Rule (aka "the angle in a semicircle is a right angle"), since PR is a diameter,angle PQR is 90 degrees, i.e. PQ is perpendicular to QR, so their gradients multiply to -1.Thus 8/12 * -6/(a-9) = -1, so 12(a-9) = 48, so a-9 = 4, so a=13, as required.
Step-by-step explanation:
The flight time from Boston to London is 6 hours 20 minutes. I arrive London at 4:05pm. What time did I set off from Boston
Answers
Final Answer: 9:45am
Answer:
Step-by-step explanation:
4:05pm go back 4hours and it’s 12:05, then go back another 2 hours, 10:05, then go back another 20mins and it’s 9:45am
Question 19). TPRS is a parallelogram. Identify mZR.
P
R
(4x+1)
T
(3x+4)
S
Answers
( 4x + 1 ) ° = ( 3x + 4 )°
Opposite side of parallelogram are always equal ...
4x + 1° = 3x + 4°
4x - 3x = 4° - 1°
x = 3°
The measure of angle R from the given parallelogram TPRS is 101°.
What is parallelogram?A parallelogram is a special kind of quadrilateral that is formed by parallel lines. The angle between the adjacent sides of a parallelogram may vary but the opposite sides need to be parallel for it to be a parallelogram. A quadrilateral will be a parallelogram if its opposite sides are parallel and congruent.
Given that, TPRS is a parallelogram.
Here, ∠T=(4x+1)° and ∠S=(3x+4)°
The sum of adjacent angles of a parallelogram is 180°.
∠T+∠S=180°
(4x+1)°+(3x+4)°=180°
7x+5=180
7x=175
x=175/7
x=25°
So, ∠T=(4x+1)°=4×25+1=101°
∠T=∠R=101° (Opposite angles of parallelogram are equal)
Therefore, the measure of angle R is 101°.
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I need help with these questions Pls
Answers
Fractions of smallest to largest
Smallest to largest: 1/4, 1/3, 2/5, 2/3, 3/4
Smallest to largest: 1/5, 2/7, 3/5, 5/7, 31/35
Smallest to largest: 31/30, 23/20, 6/5, 13/10, 7/5
Smallest to largest: 13/10, 13/6, 11/5, 25/6, 21/5
What is fraction?An element of a whole is a fraction. The number is mathematically expressed as just a fraction, where the numerator and denominator are split. Both are integers in a simple fraction. A simple fraction contains a fraction in either the numerator or the denominator. A fraction consists of two components. The numerator seems to be the number which goes just at top of the line.
Smallest to largest: 0.25, 0.33, 0.4, 0.67, 0.75
Smallest to largest: 0.2, 0.29, 0.6, 0.71, 0.89
Smallest to largest: 1.03, 1.15, 1.2, 1.3, 1.4
Smallest to largest: 1.3, 2.16, 2.2, 4.17, 4.2
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Question number 10 I need help with please
Answers
Consequently, when 3300 pounds are utilized, sand, gravel, and cement weigh 1800, 900, and 600 pounds, respectively.
what is ratio ?In algebra, ratios display how infrequently one figure is contained in another. For instance, if there are 8- oranges and 6 citrus in a fruit dish, the percentage of oranges to oranges is 8 to 6. In a similar vein, lemons to whole fruit ratio is 8 to 1, whereas lemons to oranges rate is 6 to 8. A ratio is a non-zero ordered pair of numbers, x and y b, that is stated as a / b. A ratio is really an equation that joins two ratio. The ratio can be written as 1:3, signifying that if there 1 boy and (3) girls (she has 3 girls for every boy), 3/4 of the population is female and 1/4 is male.
given
the ratio of sans , gravel and sand is 6:3:2
total pounds needed = 3300
so 6x + 3x + 2x = 3300
11x = 3300
x = 300
6x = 6 * 300 = 1800
3x = 3* 300 = 900
2x = 2* 300 = 600
Consequently, when 3300 pounds are utilized, sand, gravel, and cement weigh 1800, 900, and 600 pounds, respectively.
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Find the value of Y.
Answers
The value of y is 13√2 (optionB)
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.
Sin(tetha) = opp/hyp
cos (tetha) = adj/hyp
tan(tetha) = opp/adj
therefore cos 45 = 13/y
1/√2 = 13/y
y = 13√2
therefore the value of y is 13√2
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What is the scale factor, without units, of 1 in : 14 ft and 1 in : 30 yard
Answers
The scale factor for the ratios of 1 in : 14 feet and 1 in : 30 yards without units are 1: 168 and 1: 1080 respectively.
What is scale factorScale factor is the ratio between the scale of a given original object and a new object, which is its representation but of a different size either bigger or smaller.
For the ratio 1 in : 14 feet, we convert to the same unit in inches:
1 foot = 12 inches
14 feet = 12 × 14 inches
14 feet = 168 inches
1 in = 168 in
1 : 168
For the ratio 1 in : 30 yards, we convert to the same unit in inches:
1 yard = 36 inches
30 yards = 36 × 30 inches
30 yards = 1080 inches
1 in = 1080 in
1 : 1080
Therefore, the scale factor for the ratios of 1 in : 14 feet and 1 in : 30 yards without units are 1 : 168 and 1 : 1080 respectively.
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Convert the number to scientific notation.
120,000,000,000,000
Answers
Answer:
1.2 x [tex]10^{14}[/tex]
Step-by-step explanation:
Scientific notation OR standard form:
This is expressed in powers of 10 (i.e [tex]10^{x}[/tex])
If the decimal point cannot be visibly seen, you can deduce that the decimal point is currently located after the last digit (i.e ‘0’).
For the number to be expressed in scientific notation, the decimal point has to be placed right after the first digit (which is NOT a zero) in the left most direction.
For this number, the decimal point has to be placed right after the ‘1’. Therefore, the decimal point will be moving 14 places to the left. In addition, this 14 is considered as the positive power of 10.
Answer:
1.2 × 10¹⁴
Step-by-step explanation:
Scientific notation is written in the form [tex]a \times 10^n[/tex], where [tex]1 \leq a < 10[/tex] and [tex]n[/tex] is any positive or negative whole number.
To convert a number into scientific notation, move the decimal point to the left or right until there is one non-zero digit to the left of the decimal point.
The number of times you have moved the decimal point is the power of 10 (the value of n).
If the decimal point has moved to the left, n is positive.If the decimal point has moved to the right, n is negative.Therefore, to convert 120,000,000,000,000 to scientific notation, move the decimal point 14 places to the left so that [tex]a[/tex] is 1.2.
As the decimal point has been moved 14 places the left, we need to multiply 1.2 by 10¹⁴.
Therefore, 120,000,000,000,000 = 1.2 × 10¹⁴.
The diagonals of EFGH intersect at point M. if HM =37
Answers
Answer:
If HM = 37, then ML = 37 as the diagonals of EFGH intersect at point M.
Step-by-step explanation:
To solve the diagonals of EFGH intersecting at point M, if HM = 37, first graph the points on a coordinate grid and draw the diagonals. Then use the Pythagorean Theorem to calculate the length of HM. Finally, use trigonometry to calculate the angle measure of FHG.
Tanner is 2 years younger than his brother. Tanners age t in years is 2 years less than his brothers age b
Answers
Answer:is his brother 4 or 8
Step-by-step explanation:
I don't know for sure
The relationship between Tanner's age and his brother's age can be represented by the equation:
t = b - 2
This equation states that Tanner's age (t) is equal to his brother's age (b) minus 2 years.
How much has the value of y changed between the points (-8,4) and (8,-4)?
Answers
Look at the y value only (x,y).
4 and -4. That’s a total of 8 units down
Suppose you purchased a shirt for $16.51 using a coupon. If the shirt originally costs $25.40, what was the percent discount from the coupon? Simplify your answer completely.
Answers
The discount can be found by subtracting the sale price from the original price:
$25.40 - $16.51 = $8.89
So the discount was $8.89.
To find the percent discount, we can divide the discount by the original price and multiply by 100:
$8.89 / $25.40 * 100% = 35%
Therefore, the percent discount from the coupon was 35%.
Find (fog)(-4)!!!!!!!
Answers
Answer:
23
Step-by-step explanation:
We can think of (f o g)(-4) as f(g(-4)).
As you can see, we start with the inner function and the output of the inner function becomes the output of the outer function.
So we'll plug in -4 for x in g(x) and then the output of this becomes the input of f(x):
[tex]f(x)=2x-1\\g(x)=x^2-4\\f(g(-4))\\\\g(-4)=(-4)^2-4=12\\\\f(12)=2(12)-1=23[/tex]
An obtuse angle has a measure of 150°. The angle was bisected. Each of those angles was then trisected. Finally, each of those angles was divided into 5 equal parts. How large is the smallest angle?
Answers
The smallest angle is 5° when each of those angles was divided into 5 equal parts.
What is an Angle ?In Geometry, an angle is a figure which is produced by two rays, which share a common point called a vertex. The two rays stand in for the angle's two sides. The Roman word "Angulus," which means "corner," is where the term "angle" originates. The intersection of two curves in this plane results in the angles. Angles of several kinds can be generated on a planar surface. As follows:
Acute Angle
Obtuse Angle
Right Angle
Reflex Angle
Straight Angle
Full Angle
Obtuse Angle :An angle greater than 90° but less than 180° is referred to as a "obtuse angle." An obtuse angle is between a right angle and a straight angle.
Now in this question,
Starting with an obtuse angle of 150°, bisecting it gives two angles of 75° each.
Trisecting each of these angles gives three angles of 25° each.
Dividing each of these angles into 5 equal parts gives angles of 5° each.
Therefore, the smallest angle is 5°.
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Sharks can swim 45 miles per hour. How long would it take for the shark to swim 1 mile?
Answers
Answer:
Below
Step-by-step explanation:
distance / rate = time
1 mi / (45 mi/hr) = 1/45 hr = .02222 hr = 1 1/3 min
11.3 min
...............
A $9070 investment earns interest at
3.3% per year compounded
semiannually over 17 years. Use the
compound interest formula to
calculate the value of this investment
to the nearest cent.
Answers
Answer:
The investment will be worth $16898.67 after 17 years, compounded semiannually at 3.3% per year.
Step-by-step explanation:
The compound interest formula is:
A = P * (1 + (r/n))^(nt)
where
A is the amount of the investment after t years,
P is the principal amount (9070),
r is the annual interest rate (3.3%),
n is the number of times the interest is compounded per year (2),
t is the number of years (17).
Using the formula:
A = 9070 * (1 + (0.033/2))^(2 * 17)
A = 9070 * (1.0165)^34
A = 9070 * 1.882611
A = 16898.67 to the nearest cent.
So the investment will be worth $16898.67 after 17 years, compounded semiannually at 3.3% per year.
Answer: $3660.96 or $3661.0
Step-by-step explanation:
equation for (3,3) and (3,-3)
Answers
Answer:
x = 3
Step-by-step explanation:
A vertical line is drawn on a graph paper, which passes through points (3, 3) and (3, -3). For a vertical line, slope or gradient is undefined.
Therefore, the equation of vertical line is the common x-coordinate of the points along the line. In this case, the common x-coordinate is 3
∴Equation of line
x = 3
Help please! Easy points!
Answers
angle ACB = 80
Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown below.
Answers
The features of the sine function in this problem are given as follows:
Amplitude: 2.Midline: y = -4.Period: 3 units.The equation is given as follows:
y = 2sin(2πx/3) - 4.
How to define the sine function?The standard definition of the sine function is given as follows:
y = Asin(B(x - C)) + D.
For which the parameters are given as follows:
A: amplitude.B: the period is 2π/B.C: phase shift.D: vertical shift.The function has a minimum value of -6 and a maximum value of -2, for a difference of 4, hence the amplitude is given as follows:
2A = 4.
A = 2.
Without vertical shift, a function with amplitude of 2 would oscillate between -2 and 2, while this one oscillates between -6 and -2, hence the vertical shift is given as follows:
D = -4.
The shortest distance between repetitions can be given by 2 - (-1) = 3, hence the period is of 3 units and the coefficient B is given as follows:
2π/B = 3
B = 2π/3.
The function is at it's midline at the origin, hence it has no phase shift, and thus the equation is given as follows:
y = 2sin(2πx/3) - 4.
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The ratio of cats to dogs is 4:5 if there are 20 dogs how many are there ?
SHOW YOUR WORKG!
Answers
Answer: 16 cats
Step-by-step explanation: how many times does 5 go into 20?? 4 times (5x4) so you multiply 4 by 4
Edna has been keeping an eye on an emerald ring she likes. She just noticed that it has been marked down to $28,394.70, which is 27.1% less than its original price. What was the original price of the ring?
Round your answer to the nearest cent.
Answers
Answer:
36089.66
Step-by-step explanation:
First you have to change the percent to a decimal (0.271) then you multiply the number by $28,394.70 and get (7,694.96) last you add ($7,694.96)+ ($28,394.70) to get your answer
what is 1777771 plus 8999907
Answers
Answer: 10777678
ywyw
A particular fruit's weights are normally distributed, with a mean of 796 grams and a standard deviation of 28 grams. The heaviest 2% of fruits weigh more than how many grams?
Answer =
(Give your answer to the nearest gram.)
Answers
Answer:
325888899998898897879
Eli and Pierre wanted to buy $40 of breakfast rolls for guests. Bagels B are $1.25, and Crossiants C are $2 each. With the extra guests coming, they later increased their budget to $55.
The difference between the vertical intercepts is ________
Answers
Answer:
The difference between the vertical intercept is a ratio of 20$ so the difference in intercepts is about 2.50$ or 20 spaces.
Hope this helps
If h(x) =x+2 , g(x) =x² and f(x) =-x , find the values of x for which gh(x) =f(x)
Answers
The values of x for which gh(x) = f(x) are -1 and -4.
As a result, we have gh(x) = g(h(x)) = g(x + 2). ² = x² + 4x + 4 f(x) = -x
The equation must be solved in order to determine the range of x values for which gh(x) = f(x):
x² + 4x + 4 = -x
When all the phrases are moved to the left, we obtain:
x² + 5x + 4 = 0
This quadratic equation has the following factors:
(x + 1)(x + 4) = 0
Thus, either x + 1 = 0 or x + 4 = 0 is the answer.
We obtain x = -1 or x = -4 after solving for x.
Hence, -1 and -4 are the values of x for which gh(x) = f(x).
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