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:
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.
Related Questions
Think about the proportion and the cross products for this problem: Shelley sold one customer 5 peanut butter biscuits for $3. She sold another customer 7 beef treats for $4.20. Because the cross products are equal, which statements are true? Select all that apply.
Answers
3×4·20 equals to 12.6 please thats youre awnser
Answer:
The ratios are equivalent.
The ratios are a proportion.
Step-by-step explanation:
Determine the intercepts of the line.
y - 4= 7(x – 6)
x-intercept:
y-intercept:
Answers
Y-intercept; (0,-38)
i hope this helppppsss
PLEASE ANSWER QUICKLY ASAP
READ THE QUESTIONS CAREFULLY
Answers
Answer:
I hope it helps.
The solutions in photo ^_^
Evaluate 0.6721 x 0.0261 and express your answer in standard form
Answers
Answer:
Step-by-step explanation:
0.6821 multiplied by 0.0261 is .01754181
putting that into standard from would be 1.754181 x 10^-2
Please help ASAP. The question is down below.
Answers
Answer:
Question 1.
Option A: 2m
Question 2
Option D: (1, 1) minimum point
Step-by-step explanation:
Question 1.
Let the original length of the garden (before expansion) be = x
The new length of the garden will be x + 10m
Recall that the garden has a square geometry. That means that its area is obtained by squaring any of its sides.
This means that [tex](x +10)^2 = 144[/tex]
We can now solve for x
[tex](x +10)^2 = 144\\x^2 +20x +100 = 144\\x^2 + 20x =44\\x^2 + 20x - 44=0\\x = 2 or -22[/tex]
x cannot be a negative number, so the original length of a side of the garden is 2m. Option A
Question 2:
The coordinates of the vertex of the graph (turning point) are (x, y) [1,1]
To know whether it is a minimum or maximum point, we will have to check the coefficient of [tex]x^2[/tex] in the equation [tex]y = x^2-2x+2[/tex]
The coefficient of [tex]x^2[/tex] in the equation is 1. (If no number is present, just know that the coefficient is a one).
If the coefficient is positive, then the point is a minimum point. However, if it is negative, then the point is a maximum point.
Our coefficient is positive hence, the graph has a minimum point.
Maria had 600 coins in her wallet, but now she has 30 left of coins. What percentage of the 600 money does he have left?
Answers
Answer:5
Step-by-step explanation:30/600×100
Answer:
5%Step-by-step explanation:
[tex]\frac{30}{600} \times 100\\\\= \frac{3000}{600}\\ \\= 5\%[/tex]
What is the circumference of the circle whose equation is (x − 9)2 + (y − 3)2 = 64?
Answers
Answer:
C = 16pi or ≈50.24 units
Step-by-step explanation:
(x − 9)^2 + (y − 3)^2 = 64
This equation is in the form
(x-h)^2 + (y-k)^2 = r^2
The radius is 8
The circumference is given by
C = 2*pi*r
C = 2*pi*8
C = 16 pi
Using 3.14 as an approximation for pi
C = 16(3.14)
C ≈50.24
help me solve please
Answers
Answer:
B
Step-by-step explanation:
The side you have drawn in is 4√3 (calculate via pythagoras as √(8²-4²) = √48 = √16·3 = √4²·3 = 4√3)
So the area of the small triangle is 4*2√3 and the area of the small rectangle is 2*4√3. Together makes 4*2√3 + 2*4√3 = 16√3
Find the height of the triangle
Answers
Answer:
The height is 12 cm
Step-by-step explanation:
Hi there!
We are given a triangle with the 3 sides marked as 15, 20, and 25 and we want to find the height of it (marked as x in the problem).
This problem can seem a bit difficult, but let's see if the triangle is a right triangle first off.
One way to figure out if it is a right triangle is to apply the converse of the Pythagorean theorem.
Let's label the sides, where a is the shortest side, b is the second shortest side, and c is the longest side:
a=15
b=20
c=25
Now square a and b, then add the result together. If it's the same as c squared, then the triangle is a right triangle
15²+20²=25²
225+400=625
625=625
So we can safely say that the triangle is a right triangle
This makes the problem way easier, as there are 2 ways to find the area of a right triangle:
The first way is to multiply the legs (the sides that make up the right angle) together, then divide the result by 2
The other way is to multiply the height and the hypotenuse (the side OPPOSITE to the right angle) together, and then divide the result by 2
First, we need to figure out which sides are the legs, and which side is the hypotenuse
By the triangle inequality theorem, the hypotenuse of a right triangle is the longest side, which means that the 25 cm side is the hypotenuse, and that leaves 15 cm and the 20 cm sides as the legs
So let's find the area of the triangle using the legs
A=[tex]\frac{15*20}{2}[/tex]=[tex]\frac{300}{2}[/tex]=150
So the area of the triangle is 150 cm²
However, as mentioned above, we can also find the area of the triangle by multiplying the hypotenuse by the base, then dividing the result by 2
Which means that the area is also:
A=[tex]\frac{25x}{2}[/tex] cm²
As these both equal the area of the triangle, we can set them equal to each other. This is possible via a property known as transitivity (if a=b and b=c, then a=c)
[tex]\frac{25x}{2}=150[/tex]
Multiply both sides by 2
25x=300
Divide both sides by 25
x=12 cm
So the height of the triangle is 12 cm
Hope this helps!
ANYONE PLEASE HEEEEEEELLLLLLLPPPPPP!!!!! WILL GIVE BRAINLIEST!!!
Question 4(Multiple Choice Worth 1 points) (06.04 LC) Choose the correct simplification of the expression −3x2(5x − 4x2 − 6). −7x4 + 2x3 − 9x2 12x4 − 15x3 + 18x2 12x4 + 15x3 + 18x2 −12x4 + 15x3 − 18x2
Answers
Answer:
12x⁴ - 15x³ + 18x²
Step-by-step explanation:
1. distribute -3x²
-3x² • 5x = -15x³
-3x² • -4x² = 12x⁴
-3x² • -6 = 18x²
2. combine terms and put in descending order (standard form)
12x⁴ - 15x³ + 18x²
Answer:
12x⁴ - 15x³ + 18x²
Step-by-step explanation:
1. distribute -3x²
-3x² • 5x = -15x³
-3x² • -4x² = 12x⁴
-3x² • -6 = 18x²
2. combine terms and put in descending order (standard form)
12x⁴ - 15x³ + 18x² correct answer would be this
Find the Perimeter of the figure below, composed of a rectangle and a semicircle. Round to the nearest tenths place.
Answers
Answer:
Does the answer help you?
Consecutive terms are __________________.
Answers
Answer:
Consecutive terms are numbers which follow each other in order, without gaps, from smallest to largest.
Example:
12, 13, 14 and 15
or
2, 24, 26, 28 and 30 (Even Consecutive)
or
40, 45, 50 and 55 (Consecutive Multiples of 5)
SOURCE/GOOGLE
Answer:
Consecutive numbers are numbers that follows one other in their counting order. Each number in the order will be higher than the previous number by 1.
They are always ordered from the smallest to the largest value.
Step-by-step explanation:
Example :
7, 8, 9, 10 are consecutive numbers.
23, 24, 28, 29 are not consecutive numbers.
Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions: 2ab-6a+5b+___
Answers
Answer:
-15
Step-by-step explanation:
We proceed as follows;
In this question, we want to fill in the blank so that we can have the resulting expression expressed as the product of two different linear expressions.
Now, what to do here is that, when we factor the first two expressions, we need the same kind of expression to be present in the second bracket.
Thus, we have;
2a(b-3) + 5b + _
Now, putting -15 will give us the same expression in the first bracket and this gives us the following;
2a(b-3) + 5b-15
2a(b-3) + 5(b-3)
So we can have ; (2a+5)(b-3)
Hence the constant used is -15
90 to the nearest tenth
Answers
Answer:
90
Step-by-step explanation:
Hey there!
Well 90.000 to the nearest tenth is just 90 because there is no decimal places to round.
Hope this helps :)
Answer:
90
Step-by-step explanation:
90 has 0 ones so 90 is just 90, but rounded...
A retailer bought 1000 glass tumblers at rs 80 each.50 glass tumblers were broken and he sold the rest at Rs 96 each .find his profit or loss percent
Answers
Step-by-step explanation:
Soln
Given
Cp of 1 glass tumbler=80.1000
=Rs 80000
Number of broken glass tumbler=50
Number of remaining glass tumbler=1000-50
=950
Sp of one glass tumbler=96
Sp of 950 glass tumbler=96.950
=91200
Now
Sp > Cp so it is profit
Profit%=sp-cp/cp.100
=91200-80000/80000.100
=11200/80000.100
=14%
Answer:
14%profit
Step-by-step explanation:
Cost of 1000 tumblers=1000×80=80,000
50 of them were broken: 1000-50=950
: He sells 950 tumblers of rs.96 each= ( 950×96)
S.P=91200
We can clearly see he gains for rs.11200.
P%=gain/C.P×100
11200×80000×100
=14%
What is the area of a circle with a radius of 35 inches?
in 2
(Use 3.14 for Pi.)
Answers
Answer:
3846.5 in.
Step-by-step explanation:
πr² = (3.14)(35)² = 3846.5 in.
The value root of 46 of lies between which two consecutive integers
Answers
Answer:
6 and 7
Step-by-step explanation:
Select the correct answer. The velocity of a train relative to the ground is represented by the distance from A to B in the diagram. The velocity of a ball thrown inside the train at an angle of 66° relative to the train is represented by the distance from B to C. What is the distance from A to C (the velocity of the ball relative to the ground), correct to two decimal places? Assume that all the points in the diagram lie in the same plane. A. 21.14 m/s B. 18.03 m/s C. 17.20 m/s D. 15.00 m/s
Answers
Answer:
Step-by-step explanation:
we use cosine formula
2×15×10×cos(180-66)=15²+10²-AC²
-300 cos 66=225+100-AC²
AC²=325+300 cos 66
[tex]AC=\sqrt{325+300 cos 66} \approx 21.14 ~m/s[/tex]
The distance from A to C is 21.14. The correct option is A.
What is trigonometry?Trigonometry is the branch of mathematics which set up a relationship between the sides and angles of right-angle triangles.
Velocity is defined as the ratio of the distance moved by the object at a particular time. The velocity is a vector quantity so it needs both the magnitude and the direction.
Given that the velocity of a train relative to the ground is represented by the distance from A to B in the diagram. The velocity of a ball thrown inside the train at an angle of 66° relative to the train is represented by the distance from B to C.
The distance A to C will be calculated as,
2×15×10×cos(180-66)=15²+10²-AC²
-300 cos 66=225+100-AC²
AC=√(325+300 cos 66)
AC = 21.17 m/s
Therefore, the distance from A to C is 21.14. The correct option is A.
To know more about trigonometry follow
https://brainly.com/question/14931414
#SPJ3
WILL GIVE BRAINLEST!Two machines are printing shirts. The first machine can produce 2.400 shirts in 150 minutes. The second machine can produce 2.400 shirts in 120minutes. If the two work together, how long will it take them to produce 6,480 shirts? If necessary, round your answer to the nearest whole number.
Answers
Step-by-step explanation:
see the answer in attached picture
pls mark me as brainliest pls
Answer:
If two machine works together it takes 180 minute to complete 6480 shirts
Step-by-step explanation:
2400 ÷ 150 = 6
So here,
1st machine produces 20 shirts per minutes
let, x=minutes for shirt's production
Now,
(16+20)x=6480
36x=6480
x=6480/36
Therefor, x=180 minutes
Just need the final answer
Answers
Answer:
[tex]\boxed{\sf a) \ 15.9}[/tex]
Step-by-step explanation:
The triangle is a right triangle.
We can use trig functions to solve for x.
cos θ = adj/hyp
cos(51) = 10/x
x = 10/cos(51)
x = 15.89015729
A book contains 27 pages with print, 8 pages with print and pictures, and 3 blank
pages. What is the theoretical probability of NOT opening the book to a printed
page only?
P(NOT printed page) =
Answers
Answer:
P(NOT printed page) = 3/38
Step-by-step explanation:
Total pages = 27+8+3=38
Types of pages that satisfy requirement(s)= 3
Therfore, P(NOT printed page) = 3/38
7.
Find the number of real number solutions for the equation. x2 – 18 = 0
A. cannot be determined
B. 0
C. 1
D. 2
Answers
The solutions are sqrt(18) and -sqrt(18)
Coherence
5. Simon's teacher asked him to e-mail her a copy of the outline for his essay on American
History: When drafting the e-mail, what level of diction should Simon use?
informal
formal
standard
foundational
Answers
Answer:
The correct option is;
Formal
Step-by-step explanation:
The common levels of diction are formal, informal, and popular, with formal diction being the most selective of the word choices
Formal diction is used when when communicating in a situation that is formal
Formal diction uses languages that is devoid of slang and grammatically correct
Formal language is precise, grammatically correct language that does not use slang used in communication for legal, professional, business and academic purposes.
6a+2b-6c+4 if a=5,b=3,and c=-1
Answers
Answer:
46
Step-by-step explanation:
First, start off by substituting the values of a, b, and c, into the equation.
We know a = 5, b = 3, and c = -1, so now substitute.
6a + 2b - 6c + 4
6(5) + 2(3) - 6(-1) + 4
Now that we've substituted the values, we can solve the equation.
6(5) + 2(3) - 6(-1) + 4
30 + 6 + 6 + 4
= 46
So, the answer is 46.
I hope this helps! ôヮô
Answer:
46
Step-by-step explanation:
All you need to do is subsitute the variable with what it says the number is.
Let's first start out with 6a from the equation. We know that a= 5, so that means it wants you to multiply 6 by 5, which is 30.
Now let's do 2b. It says b= 3, so do 2 times 3. It equals 6.
So far we have 30+6-6c+4
Let's do -6c. We know that c= -1, so let's multiply -1 and -6. Negative and negative equals positive. And 6 times 1 is 6. So the outcome is positive 6.
We got 30+6+6+4
Solve.
30+6= 36
36+6= 42
And 42+4= 46. :)
A delivery truck company just bought a new delivery truck and they need to know the maximum volume it can carry. In the front of the truck, there is an extra ledge that sticks out over the driver's cab for extra storage space. What is the maximum amount of cargo that can fit into the new truck?
Answers
Answer:
The answer is below
Step-by-step explanation:
To find the maximum amount of cargo the truck can carry, we need to find the volume of the truck.
Volume = length × width × height.
Firstly 1 feet (1') = 12 inches (12"),
For the extra ledge that sticks out, the height = 7'8" = 7.667 feet, the width = 16'9" - 14'3" = 16.75 - 14.25 = 2.5 feet, the length = 2'7" = 2.583 feet
Volume of extra ledge = length × width × height = 2.583 × 2.5 × 7.667 = 49.5 feet³
For the truck, the height = 7'8" = 7.667 feet, the length = 14'3" = 14.25 feet, the width = 6'6" = 6.5 feet
Volume of truck = length × width × height = 14.25 × 6.5 × 7.667 = 710.16 feet³
The maximum volume = volume of extra ledge + volume of truck = 49.5 + 710.16 = 759.66 feet³
Find the value of x. A. 53–√ m B. 241−−√ m C. 6 m D. 6+35–√ m
Answers
Answer:
x = 2√41 mStep-by-step explanation:
Since the triangle is a right angled triangle we can use Pythagoras theorem to find the missing side x
Using Pythagoras theorem we have
a² = b² + c²
where a is the hypotenuse
From the question x is the hypotenuse
So we have
[tex] {x}^{2} = {8}^{2} + {10}^{2} [/tex][tex] {x}^{2} = 64 + 100[/tex][tex] {x}^{2} = 164[/tex]Find the square root of both sides
We have the final answer as
x = 2√41 mHope this helps you
Answer:
2 sqrt(41) =c
Step-by-step explanation:
Since this is a right triangle, we can use the Pythagorean theorem
a^2 + b^2 = c^2
8^2 + 10^2 = c^2
64+ 100 = c^2
164 = c^2
take the square root of each side
sqrt(164) = sqrt(c^2)
sqrt(4*41) = c
2 sqrt(41) =c
Which equation represents a proportional relationship between
x and y?
A: k/x = y
B: k - x = y
C: k + x = y
D: k * x = y
Answers
Answer:
D: k*x=y
Step-by-step explanation:
a proportional relationship is basically a linear equation (y=mx+b)
The area of a square of side X is 8. What's the area of a square of side 3x
Answers
Answer:
Step-by-step explanation:
We know, for square
[tex](side)^2=(Area)\\=>X^2=8\\[/tex]
∴[tex]X=2\sqrt{2}[/tex] unit
[tex]Now, Side=3X[/tex]
[tex]Then,Area=(3X)^2=(3*2\sqrt{2} )^2=(6\sqrt{2} )^2=72 sq. unit[/tex]
hope you have understood this...
pls mark my answer as the brainliest
The area of square of side 3X is 72 square unit.
What is square?The square is a 4 sided figure, each side of the square is equal and make a right angle.
The area of square having sided a unit can be given by a² square unit.
Given that,
Area of square having side X is 8.
Since, formula for area of square having side X is X².
Implies that,
X² = 8
X = 2√2
The area of square having side 3X
side =3 × 2√2
side = 6√2
The area of square = (6√2)²
= 36 x 2
= 72
The area of square is 72 square unit.
To know more about Square on:
https://brainly.com/question/28776767
#SPJ2
Find the distance between the points A(13, 2) and B(7, 10). The distance between the two points is
Answers
Answer:
The answer is 10 unitsStep-by-step explanation:
The distance between two points can be found by using the formula
[tex]d = \sqrt{( {x1 - x2})^{2} + ({y1 - y2})^{2} } [/tex]where
(x1 , y1) and (x2 , y2) are the points
From the question , the points are
A(13, 2) and B(7, 10)
The distance between them is
[tex] |AB| = \sqrt{( {13 - 7})^{2} + ({2 - 10})^{2} } \\ = \sqrt{ {6}^{2} + ( { - 8})^{2} } \\ = \sqrt{36 + 64} \\ = \sqrt{100} [/tex]We have the final answer as
10 unitsHope this helps you
On a separate piece of graph paper, graph y=-Ixl +3 then click on the graph until the correct one appears.
Answers
Answer:
That is the graph for y = -|x| + 3