Calculus!

The volume of a substance, A, measured in cubic centimeters increases according to the exponential growth model dA/dt = 0.3A, where t is measured in hours. The volume of another substance, B, also measured in cubic centimeters increases at a constant rate of 1 cm^3 per hour according to the linear model dB/dt = 1. At t = 0, substance A has a volume A(0) = 3 and substance B has size B(0) = 5. At what time will both substances have the same volume?

Would it be correct to write the growth model of substance B as x + 5? And how could I write the growth model of substance A? Thank you in advance, and sorry for the poor formatting.

Answers

Answer 1

Answer:

The two substances will have the same volume after approximately 3.453 hours.

Step-by-step explanation:

The volume of substance A (measured in cubic centimeters) increases at a rate represented by the equation:

[tex]\displaystyle \frac{dA}{dt} = 0.3 A[/tex]

Where t is measured in hours.

And substance B is represented by the equation:

[tex]\displaystyle \frac{dB}{dt} = 1[/tex]

We are also given that at t = 0, A(0) = 3 and B(0) = 5.

And we want to find the time(s) t for which both A and B will have the same volume.  

You are correct in that B(t) is indeed t + 5. The trick here is to multiply both sides by dt. This yields:

[tex]\displaystyle dB = 1 dt[/tex]

Now, we can take the integral of both sides:

[tex]\displaystyle \int 1 \, dB = \int 1 \, dt[/tex]

Integrate. Remember the constant of integration!

[tex]\displaystyle B(t) = t + C[/tex]

Since B(0) = 5:

[tex]\displaystyle B(0) = 5 = (0) + C \Rightarrow C = 5[/tex]

Hence:

[tex]B(t) = t + 5[/tex]

We can apply the same method to substance A. This yields:

[tex]\displaystyle dA = 0.3A \, dt[/tex]

We will have to divide both sides by A:

[tex]\displaystyle \frac{1}{A}\, dA = 0.3\, dt[/tex]

Now, we can take the integral of both sides:

[tex]\displaystyle \int \frac{1}{A} \, dA = \int 0.3\, dt[/tex]

Integrate:

[tex]\displaystyle \ln|A| = 0.3 t + C[/tex]

Raise both sides to e:

[tex]\displaystyle e^{\ln |A|} = e^{0.3t + C}[/tex]

Simplify:

[tex]\displaystyle |A| = e^{0.3t} \cdot e^C = Ce^{0.3t}[/tex]

Note that since C is an arbitrary constant, e raised to C will also be an arbitrary constant.

By definition:

[tex]\displaystyle A(t) = \pm C e^{0.3t} = Ce^{0.3t}[/tex]

Since A(0) = 3:

[tex]\displaystyle A(0) = 3 = Ce^{0.3(0)} \Rightarrow C = 3[/tex]

Therefore, the growth model of substance A is:

[tex]A(t) = 3e^{0.3t}[/tex]

To find the time(s) for which both substances will have the same volume, we can set the two functions equal to each other:

[tex]\displaystyle A(t) = B(t)[/tex]

Substitute:

[tex]\displaystyle 3e^{0.3t} = t + 5[/tex]

Using a graphing calculator, we can see that the intersect twice: at t ≈ -4.131 and again at t ≈ 3.453.

Since time cannot be negative, we can ignore the first solution.

In conclusion, the two substances will have the same volume after approximately 3.453 hours.

Calculus!The Volume Of A Substance, A, Measured In Cubic Centimeters Increases According To The Exponential

Related Questions

Whats his mistake and the correct awnser

Answers

Answer:

θ ≈ 35.7°

Step-by-step explanation:

His mistake was in using the wrong trig. ratio

Instead of using cosine he should have used the sine ratio

sinθ = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{7}{12}[/tex] , then

θ = [tex]sin^{-1}[/tex] ([tex]\frac{7}{12}[/tex] ) ≈ 35.7° ( to the nearest tenth )

Which of the integers shown is the greatest?

Answers

Answer:

-3

Step-by-step explanation:

because as you vo more to the negative side the less.it gets and more to the positive side it gets moe

If p =(3x+1) and q = ( 3x-1),show that : pq+1 =x2

Answers

Answer:

See explanation

Step-by-step explanation:

It should be 9x² not x²

p = (3x + 1) and q = (3x - 1)

pq + 1 = (3x + 1)(3x - 1) + 1

= (9x² - 3x + 3x - 1) + 1

= (9x² - 1) + 1

= 9x² - 1 + 1

= 9x²

Therefore,

pq + 1 = 9x²

Can someone please help, I'll give out 20 points !

Question - Where is the blue point on the number line?

Answers

I believe it would be negative three (-4) since the lines are thinner however; there is a chance they are skip counting by 2. In that case the answer would be negative six (-6).

Answer:

-6

Step-by-step explanation:

The number line is created by skip counting of 2.

So, -2 ,  then -4 and then -6.

So,  blue dotted point is (-6)

Please help!!!!
Oak wilt is a fungal disease that infects oak trees. Scientists have discovered that a single tree in a small forest is infected with oak wilt. They determined that they can use this exponential model to predict the number of trees in the forest that will be infected after t years.
f(t) = e^0.4t
1. The scientists believe the forest will be seriously damaged when 21 or more of the forest’s 200 oak trees are infected by oat wilt. According to their model, how many years will it take for 21 of the trees to become infected?

Type the correct answer in the box. Use numerals instead of words. Round your answer to the nearest tenth.

2. Rewrite the exponential model as a logarithmic model that calculates the number of years g (x) for the number of infected trees to reach a value of x.

Answers

To solve this question, we need to solve an exponential equation, which we do applying the natural logarithm to both sides of the equation, both to find the needed time and to find the inverse function. From this, we get that:

It will take 7.6 years for 21 of the trees to become infected.The logarithmic model is: [tex]g(x) = \frac{\ln{x}}{0.4}[/tex]

Number of trees infected after t years:

The number of trees infected after t years is given by:

[tex]f(t) = e^{0.4t}[/tex]

Question 1:

We have to find the number of years it takes to have 21 trees infected, that is, t for which:

[tex]f(t) = 21[/tex]

Thus:

[tex]e^{0.4t} = 21[/tex]

To isolate t, we apply the natural logarithm to both sides of the equation, and thus:

[tex]\ln{e^{0.4t}} = \ln{21}[/tex]

[tex]0.4t = \ln{21}[/tex]

[tex]t = \frac{\ln{21}}{0.4}[/tex]

[tex]t = 7.6[/tex]

Thus, it will take 7.6 years for 21 of the trees to become infected.

Question 2:

We have to find the inverse function, that is, first we exchange y and x, then isolate x. So

[tex]f(x) = y = e^{0.4x}[/tex]

[tex]e^{0.4y} = x[/tex]

Again, we apply the natural logarithm to both sides of the equation, so:

[tex]\ln{e^{0.4y}} = \ln{x}[/tex]

[tex]0.4y = \ln{x}[/tex]

[tex]g(x) = \frac{\ln{x}}{0.4}[/tex]

Thus, the logarithmic model is:

[tex]g(x) = \frac{\ln{x}}{0.4}[/tex]

For an example of a problem that uses exponential functions and logarithms, you can take a look at https://brainly.com/question/13812761

Fine FG , given that line HF is perpendicular bisector or EG

Answers

Answer:

FG = 7

Step-by-step explanation:

We'll begin by calculating HF. This can be obtained by using the pythagoras theory as illustrated below:

EF = 7

EH = 3

HF =?

EF² = EH² + HF²

7² = 3² + HF²

49 = 9 + HF²

Collect like terms

49 – 9 = HF²

40 = HF²

Take the square root of both side

HF = √40

Finally, we shall determine FG. This can be obtained as follow:

GH = 3

HF = √40

FG =.?

FG² = GH² + HF²

FG² = 3² + (√40)²

FG² = 9 + 40

FG² = 49

Take the square root of both side

FG = √49

FG = 7


use a double angle or half angle identity to find the exact value of each expression

Answers

θ is given to be in the fourth quadrant (270° < θ < 360°) for which sin(θ) < 0 and cos(θ) > 0. This means

cos²(θ) + sin²(θ) = 1   ==>   sin(θ) = -√[1 - cos²(θ)] = -3/5

Now recall the double angle identity for sine:

sin(2θ) = 2 sin(θ) cos(θ)

==>   sin(2θ) = 2 (-3/5) (4/5) = -24/25

Answer the questions

Answers

Answer:

c) x= 15

d) x= 5

Step-by-step explanation:

c)4x + 5+ 7x +10=180

solve it like usually

for d) 3x + 15+12x=90

Answer:

15 and 3

Step-by-step explanation:

the first figure total sum of angles is 180 since it is a straight line

the second figure is a right ane(90) so sum of angles is 90

Part B
Question
Type the correct answer in the box.
The y-intercept of the parent function, f(x), is
I
The y-intercept of the child function, g(x), is

Answers

Answer:

The y-intercept of the parent function, f(x), is 0

The y-intercept of the child function, g(x), is  7

Step-by-step explanation:

Since the parent function passes through the origin, its y-intercept is 0. Once the function is translated 7 units up, its y-intercept is 7.

The solution is :

Vertex is (-2, 0)

y-intercept is 4.

What is vertex?

In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.

here, we have,

to find the function translated 2 units left, we just need to substitute the 'x' in the equation by 'x+2':

f(x) = (x+2)^2 = x2 + 4x + 4

Then, to find the vertex, we can use the formula:

x_v = -b / 2a

x_v = -4 / 2 = -2

Now, finding the y_vertex, we have:

f(x_v) = (-2)^2 + 4*(-2) + 4 = 0

So the vertex is (-2, 0)

To find the y-intercept, we make x = 0 and then find f(x):

f(0) = 0 + 0 + 4 = 4

So the y-intercept is 4.

To learn more on vertex click:

brainly.com/question/12563262

#SPJ2

complete question:

Identify the vertex and y-intercept of the graph of the function translated 2 units left from the parent function f(x) = x².

vertex: ( ? , ? )

y-intercept: ?

how do i find upper and lower quartile
please help will give brainliest

Answers

Answer:

Step-by-step explanation:

Arrange the data set in ascending order. then find the median.

Data in ascending order:

2,  5 , 6 , 7 , 8 , 11 , 15

Median = 7

Lower quartile:

Then  again find the median of the numbers from the first to the number before the median. This is the lower quartile.

2 , 5, 6

Median = 5  ----> this  is the lower quartile.

Upper quartile:

Then  again find the median of the numbers that comes after the median to the end of the data set. This is the upper quartile.

8 , 11 , 15

Median = 11 ------> is the upper quatile

i will brainliest
avoid spam

Express each of the following percentages as a fraction and simplify it.

(i) 25% (ii) 40% (iii) 16% (iv) 150%

(v) 120% (vi) 58% (vii) 32% (viii) 175%

Answers

[tex]\boxed{\large{\bold{\textbf{\textsf{{\color{blue}{Answer}}}}}}:)}[/tex]

[tex]\sf{25\%=\dfrac{25}{100}=\dfrac{1}{4} }[/tex]

[tex]\sf{ 40\%=\dfrac{40}{100}=\dfrac{2}{5} }[/tex]

[tex]\sf{16\%=\dfrac{16}{100}=\dfrac{4}{25} }[/tex]

[tex]\sf{150\%=\dfrac{150}{100}=\dfrac{3}{2} }[/tex]

[tex]\sf{120\%=\dfrac{120}{100}=\dfrac{6}{5} }[/tex]

[tex]\sf{58\%=\dfrac{58}{100}=\dfrac{29}{50} }[/tex]

[tex]\sf{32\%=\dfrac{32}{100}=\dfrac{8}{25} }[/tex]

[tex]\sf{175\%=\dfrac{175}{100}=\dfrac{7}{4} }[/tex]

[tex]\sf{ }[/tex]

[tex]\sf{ }[/tex]

[tex]\sf{ }[/tex]

[tex]\sf{ }[/tex]

[tex]\sf{ }[/tex]

[tex]\sf{ }[/tex]

[tex]\hookrightarrow\sf{ 25\% =\dfrac{25}{100}=\boxed{\bf \dfrac{1}{4} }} \\\hookrightarrow\sf{ 40\% =\dfrac{40}{100}=\boxed{\bf \dfrac{2}{5} }} \\\hookrightarrow \sf{16\%=\dfrac{16}{100}=\boxed{\bf\dfrac{4}{25}} } \\ \hookrightarrow\sf{150\%=\dfrac{150}{100}=\boxed{\bf\dfrac{3}{2} }} \\ \hookrightarrow\sf{120\%=\dfrac{120}{100}=\boxed{\bf\dfrac{6}{5} }} \\\hookrightarrow \sf{58\%=\dfrac{58}{100}=\boxed{\bf\dfrac{29}{50}} } \\\hookrightarrow \sf{32\%=\dfrac{32}{100}=\boxed{\bf\dfrac{8}{25} }} \\ \hookrightarrow\sf{175\%=\dfrac{175}{100}=\boxed{\bf\dfrac{7}{4}} } \\ [/tex]

Please help!! Thank you!

Answers

Answer:

[tex]\text{As }x\rightarrow \infty, f(x)\rightarrow -\infty,\\\text{As }x\rightarrow -\infty, f(x)\rightarrow -\infty[/tex]

Step-by-step explanation:

This question is asking for the function's end behavior. An even degree power function with a negative leading coefficient forms a parabola concave down. This means, its vertex represents the maximum and the parabola opens downward (towards negative infinity).

Thus,

[tex]\lim_{x\rightarrow \infty}f(x)=-\infty,\\\lim_{x\rightarrow-\infty}f(x)=-\infty[/tex]

This corresponds with the first answer choice listed:

[tex]\text{As }x\rightarrow \infty, f(x)\rightarrow -\infty,\\\text{As }x\rightarrow -\infty, f(x)\rightarrow -\infty[/tex]


Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.
If x+2 is a factor of x^3-6x^2-11x+k then k=

Answers

Answer:

k=10

Step-by-step explanation:

Brainliest please~

Answer:

K=10

Step-by-step explanation:

Correct on Plato

You and a friend decide to take a road trip. You drive an average speed of 70 mph and your friend drives an average of 75 mph. You want to drive at least 500 miles per day. You also plan to spend no more than 10 hours driving each day. Which of the following is a system of linear inequalities that represents this situation? Let x = number of hours you drive and let y represent the number of hours your friend will drive.

Answers

Given:

your speed = 70 mph

Your friend's speed = 75 mph

You want to drive at least 500 miles per day.

You also plan to spend no more than 10 hours driving each day.

To find:

The system of linear inequalities that represents this situation.

Solution:

Let x be the number of hours you drive and let y represents the number of hours your friend will drive.

You also plan to spend no more than 10 hours driving each day.

[tex]x+y\leq 10[/tex]

Your speed is 70 mph and your friend's speed is 75 mph. So, the distance covered in x and y hours are 70x miles and 75y miles respectively.

You want to drive at least 500 miles per day. So, the total distance must be greater than or equal to 500.

[tex]70x+75y\geq 500[/tex]

[tex]5(14x+15y)\geq 500[/tex]

Divide both sides by 5.

[tex]14x+15y\geq 100[/tex]

Therefore, the required system of inequalities has two inequalities [tex]x+y\leq 10[/tex] and [tex]14x+15y\geq 100[/tex].

No idea how to do

[tex]\frac{4r + 20}{r + 5}[/tex]

when r ≠ 5 ?

Answers

Factorize the numerator:

4r + 20 = 4r + 4×5 = 4 (r + 5)

I think you meant to say r ≠ -5, which means r + 5 ≠ 0, so that the denominator is never zero and so the expression is defined (no division by zero). This lets you cancel the factor of r + 5 in the numerator with the one in the denominator:

(4r + 20)/(r + 5) = 4 (r + 5)/(r + 5) = 4

Lines 3x-2y+7=0 and 6x+ay-18=0 is perpendicular. What is the value of a?

1/9
9
-9
-1/9

Pls answer​

Answers

Answer:

[tex]\boxed{\sf a = 9 }[/tex]

Step-by-step explanation:

Two lines are given to us which are perpendicular to each other and we need to find out the value of a . The given equations are ,

[tex]\sf\longrightarrow 3x - 2y +7=0 [/tex]

[tex]\sf\longrightarrow 6x +ay -18 = 0 [/tex]

Step 1 : Convert the equations in slope intercept form of the line .

[tex]\sf\longrightarrow y = \dfrac{3x}{2} +\dfrac{ 7 }{2}[/tex]

and ,

[tex]\sf\longrightarrow y = -\dfrac{6x }{a}+\dfrac{18}{a} [/tex]

Step 2: Find the slope of the lines :-

Now we know that the product of slope of two perpendicular lines is -1. Therefore , from Slope Intercept Form of the line we can say that the slope of first line is ,

[tex]\sf\longrightarrow Slope_1 = \dfrac{3}{2} [/tex]

And the slope of the second line is ,

[tex]\sf\longrightarrow Slope_2 =\dfrac{-6}{a} [/tex]

Step 3: Multiply the slopes :-

[tex]\sf\longrightarrow \dfrac{3}{2}\times \dfrac{-6}{a}= -1 [/tex]

Multiply ,

[tex]\sf\longrightarrow \dfrac{-9}{a}= -1[/tex]

Multiply both sides by a ,

[tex]\sf\longrightarrow (-1)a = -9 [/tex]

Divide both sides by -1 ,

[tex]\sf\longrightarrow \boxed{\blue{\sf a = 9 }} [/tex]

Hence the value of a is 9 .

what is the solution to this equation?
5x-4+3x=36

A. x=16
B. x=5
C. x=20
D. x=4

Answers

B. x = 5

tip : if in a rush just plug in the number and see if its true

8x - 4 = 36

x = 5

When a boy stands on the bank of a river and looks across to the other bank, the angle of depression is 12°. If he climbs to the top of a 10 ft tree and looks across to other bank, the angle of depression is 15°. What is the distance from the first position of the boy to the other bank of the river? How wide is the river? Give your answers to the nearest foot.

Answers

Please look at the scanned picture.

pls answer fast i need help

Answers

Do it by your self  ok ???

The formula for the circumference of a circle is C = 21
r, where r is the radius and C is the circumference. The
equation solved for r is raza
Find the radius of a circle that has a circumference of
16.
r= 4
r= 8
r = 12
r= 16

Answers

Answer:

r=8

Step-by-step explanation:

Given sin
=
=
V23 and tan o
find cos 0.
23
11 )
12
?
COS 0 =
TO

Answers

Answer: [tex]\frac{11}{12}[/tex]

Step-by-step explanation:

Use trigonometry (shown in image below) to find the 3 side-lengths:

The side opposite of θ = [tex]\sqrt{23}[/tex]The base side adjacent to θ = 11The hypotenuse = 12

According to right triangle trigonometry:

cos θ = [tex]\frac{adjacent}{hypotenuse}[/tex]

Substitute in the values:

cos θ = [tex]\frac{11}{12}[/tex]

Dan is on his way home in his car. He has driven 10 miles so far, which is one-half of the way home. What is the total length of his drive?

Answers

Answer:

20 miles

Step-by-step explanation:

10miles × 2 = 20miles

Half of 20 equals 10,
20 miles is the total length of his drives in miles.

the functions f and g are defined as follows

Answers

Answer:

f(- 3) = 27 and g(6) = - 15

Step-by-step explanation:

Substitute x = - 3 into f(x) and x = 6 into g(x) , that is

f(- 3) = - 2(- 3)² - 3(- 3) = - 2(9) + 9 = 18 + 9 = 27

g(6) = - 2(6) - 3 = - 12 - 3 = - 15

-15 would be your answer

I hope I helped, stay safe. Xx

What is the length of side s of the square shown below? 450 90" A. 4-3 B. 1 C. 4 D. 2 E F. 2.5​

Answers

Answer:

E

Step-by-step explanation:

Using Pythagoras' identity in the right triangle

s² + s² = 2²

2s² = 4 ( divide both sides by 2 )

s² = 2 ( take the square root of both sides )

s = [tex]\sqrt{2}[/tex]

Answer:

E.

Step-by-step explanation:

we know the diagonal of the square : 2

as you can see in the picture, there is a right-angled triangle with the baseline of Hypotenuse being the diagonal, and 2 times s being the 2 sides.

that means we can use Pythagoras to calculate s :

2² = s² + s²

4 = 2s²

2 = s²

s = sqrt(2)

Choose the correct simplification of the expression b5 ⋅ b4.

b
b9
b20
b−1

Answers

Answer:

b20

Step-by-step explanation:


Expand the following:
a) 2(x+3)
b) 512x-4)
C) 4(2x + 1)
d) 6 x - 4y

Answers

Answer:

A.2x+6

B.60x-20

C.8x+4

D.6x-24y

find the value of x

Answers

4x - 7 = 3x + 9

x = 16

they're equal because they are opposites by the vertex.

hope it helps :)

find area of shaded region by using formula a^2-b^2.​

Answers

Answer:

216 cm^2

Step-by-step explanation:

15^2 -3^2 = 225 -9

= 216

In ΔFKR, which side is included between ∠F and ∠R?

Answers

Answer:

First choice

Step-by-step explanation:

It gives you the endpoints of the included side, F and R. Therefore, the side is FR or RF.

The blue dot is at what value on the number line?

Answers

Answer:

[tex]8[/tex]

Step-by-step explanation:

Hope it helps you:D

*copied*

Answer:

"8"

4,6,8

Step-by-step explanation:

Other Questions
Choose a suitable word/ phrase to fill in the sentence. "Fossil fuels are ______energy sources such as coal, fuel oil and natural gas formed from dead plants and animals underground."(1 Point)non-renewablenewablerenewableunrenewable ShabaLikeAnOobaLikeAMeebo What is 3x+8x=13+31? 3.In each box there are 22 eggs. How many eggs do I have if I have 12 boxes? You are going to travel by coach together for a whole day. There are some things you might take for the journey. Decide together what you will take The gcf of two numbers is 3 and their lcm is 180, if one of the numbers is 45 then found the second number Can someone help me on this math question its confusing ill give brainly when right!!! who is the father of computer what is the domain of the function in the graph? According to the interest parity condition, if the domestic interest rate is 12 percent and the foreign interest rate is 10 percent, then the expected _________ of the foreign currency must be _________ percent. Group of answer choices Piecewise functions alg1 One of the strengths of Six Sigma program's has been its a.laser-like focus on worker satisfaction. b.clarity of process and the steps companies must take to adopt it. c.contribution to research in production quality. d.increase in customer loyalty. What is 3.24 ( 4 repeating) as a fraction? 100 x 20 + 30 help me 21in. = _____ft. terminating or repeating Lymph ______ are found throughout the body and are a signal of a disease, infection, or problem if they become and remain swollen for long periods of time.______ are pathogens that cannot be killed with antibiotics because they are non-living. They take over cellular organelles to replicate and spread.Because they are also eukaryotes with their own cell membranes and membrane-bound organelles, ______ are often difficult to combat with medicines.A cell signal called ______ is produced by mast cells. These molecules cause more fluid to spread to an infected area._______ are the types of cells that produce antibodies.All pathogens have unique identifying molecules on their surfaces called ______. Antibodies bind to these, and targeting the pathogen to be destroyed by the immune system.______ is a non-specific response to disease, in which the body temperature rises how did Industrial Development help to create big towns and cities If ABCD is dilated by a factor of }, thecoordinate of B' would be:543B21-54 3-2-1012345-1AB' = ([?], [ ]D-3Enter which tradition of our society need to be reformed list them and write the ways to Reform them Six pairs of shoes cost as much as 1 coat, 2 pairs of jeans cost as much as 3 pairs of shoes, and 4 pairs of socks cost as much as one pair of jeans. How many coats could I exchange for 64 pairs of socks?