when a vertical beam of light passes through a transparent medium, the rate at which its intensity i decreases is proportional to i(t), where t represents the thickness of the medium (in feet). in clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity i0 of the incident beam. what is the intensity of the beam 17 feet below the surface? (give your answer in terms of i0. round any constants or coefficients to five decimal places.)
Answers
The intensity of the beam 17 feet below the surface is 0.440265 times the initial intensity i0 of the incident beam is I(17) ≈ 0.002678.
It can be calculated as:
Let I(t) be the intensity of the beam at a depth of t feet below the surface, and
let k be a constant of proportionality.
Then we have:
\(dI/dt = -kI\)
This equation says that the rate of change of intensity with respect to depth is proportional to the intensity itself, and the negative sign indicates that intensity decreases as depth increases.
We can solve this differential equation using separation of variables:
\(dI/I = -k dt\)
\(\int\ dI/I = \int\ -k dt\)
\(ln(I) = -kt + C\)
\(I = e^{(C - kt)}\)
where C is the constant of integration.
Now we can use the given information to find the value of k and the constant of integration C.
We know that at a depth of 3 feet below the surface, the intensity is 25% of the initial intensity i0:
\(I(3) = 0.25 i0\)
\(e^{(C - 3k)} = 0.25 i0\)
We also know that the depth at which we want to find the intensity is 17 feet below the surface:
t = 17
Now we can use the equation we derived earlier to find the intensity at a depth of 17 feet:
\(I(17) = e^{(C - 17k)}\)
To find the constant of integration C and the constant of proportionality k, we can use the fact that we have two equations with two unknowns. First, we can solve the equation for C:
\(e^{(C - 3k)} = 0.25 i0\)
\(C - 3k = ln{(0.25 i0)}\)
\(C = ln{(0.25 i0)} + 3k\)
Now we can substitute this expression for C into the equation for I(17):
\(I(17) = e^{(C - 17k)}\)
\(I(17) = e^{(ln(0.25 i0) + 3k - 17k)}\)
\(I(17) = e^{(ln(0.25 i0) - 14k)}\)
Finally, we can solve for k using the fact that we know the intensity decreases by a factor of 0.25 when the depth increases from 0 to 3 feet:
\(dI/dt = -kI\)
\(ln(I) = -kt + C\)
\(I(3) = 0.25 i0\)
\(e^{(C - 3k)} = 0.25 i0\)
Taking the natural logarithm of both sides, we have:
\(C - 3k = ln{(0.25 i0)}\)
Substituting the expression for C we derived earlier, we have:
\(ln{(0.25 i0)} + 3k - 3k = ln{(0.25 i0)}\)
\(ln{(0.25 i0)} = ln{(0.25 i0)}\)
This equation is true for all values of k, so we can choose any value for k that satisfies the differential equation.
For simplicity, we can choose\(k = ln(4)/3\), which makes the constant of proportionality equal to\(-ln(4)/3.\)
Now we can substitute this value of k into our expression for I(17) and simplify:
\(I(17) = e^{(ln(0.25 i0) - 14k)}\)
\(I(17) = e^{(ln(0.25 i0) - 14ln(4)/3)}\)
\(I(17) = 0.25 i0 e^{(-14ln(4)/3)}\)
\(I(17) \approx 0.002678\)
The intensity of the beam 17 feet below the surface is approximately 0.002678.
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Related Questions
Two mechanics worked on a car. The first mechanic worked for hours, and the second mechanic worked for hours. Together they charged a total of . What was the rate charged per hour by each mechanic if the sum of the two rates was per hour?
Answers
Answer:
The rate charged by first mechanic per hour= x =$85
The rat charged by second mechanic per hour =y = $70
Step-by-step explanation:
Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 15 hours. Together they charged a total of $1900. What was the rate charged per hour by each mechanic if the sum of the two rates was $155 per hour?
Solution
Let
x= hourly rate of the first mechanic
y= hourly rate of the second mechanic
Derive two equations to solve for the two unknowns
10x + 15y = 1900 (1)
x + y = 155 (2)
From (2)
x + y = 155
x=155-y
Substitute x=155-y into (1)
10x + 15y = 1900
10(155-y) + 15y =1900
1550 -10y + 15y =1900
5y =1900-1550
5y=350
Divide both sides by 5
y= 70
Substitute y=70 into (2)
x + y = 155
x + (70) =155
x=155 - 70
= 85
x= 85
The rate charged by first mechanic per hour= x =$85
The rat charged by second mechanic per hour =y = $70
Will give brainlyest!!!!:
A construction company is planning to pour concrete for a driveway. The length of the driveway is 16 feet longer than its width w.
a. Write an expression for the area of the driveway.
b. Find the dimensions of the driveway if it has an area of 260 square feet.
Answers
The length of the driveway is 16 feet longer than its width "w".
Length is "w+16".
-----------------------------
32. Write an expression for the area of the driveway.
Area = w(w+16) = w^2+16w sq. ft.
------------------------------------------
33. Find the dimensions of the driveway if it has an area of 260 square feet.
Solve 260 = w^2+16w
w^2+16w-260 = 0
(w+26)(w-10) = 0
Positive solution:
width = 10 ft
length = 10+16 = 26 ft.
Why is 5x-15x=0 not a quadratic equation?
Answers
Answer:
because the formula for a parabola is f(x)=ax^2+bx+c
Step-by-step explanation:
Given m||n,
find the value of x and y.
Answers
Answer:
Step-by-step explanation:
4x -5 = 3x +11 corresponding angles
x= 16
y + 4x - 5 = 180
y + 4(16) = 185
y - 64 =185
y = 121
I need some help please
Answers
Answer:
3
Step-by-step explanation:
You invest $20,000 in the stock market. The stock market then plummets
over the next few weeks. Each day, your investment loses half of its value. How
much will you have invested after 14 days? Write the geometric sequence
formula and show all of your work.
Answers
After 14 days, you will have approximately $2.4414 invested in the stock market.
The amount you will have invested after 14 days can be calculated using the geometric sequence formula. The formula for the nth term of a geometric sequence is given by:
an = a1 x \(r^{(n-1)\)
Where:
an is the nth term,
a1 is the first term,
r is the common ratio, and
n is the number of terms.
In this case, the initial investment is $20,000, and each day the investment loses half of its value, which means the common ratio (r) is 1/2. We want to find the value after 14 days, so n = 14.
Substituting the given values into the formula, we have:
a14 = 20000 x\((1/2)^{(14-1)\)
a14 = 20000 x \((1/2)^{13\)
a14 = 20000 x (1/8192)
a14 ≈ 2.4414
Therefore, after 14 days, you will have approximately $2.4414 invested in the stock market.
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The amount you will have invested after 14 days is given as follows:
$2.44.
What is a geometric sequence?A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.
The explicit formula of the sequence is given as follows:
\(a_n = a_1q^{n-1}\)
In which \(a_1\) is the first term of the sequence.
The parameters for this problem are given as follows:
\(a_1 = 20000, q = 0.5\)
Hence the amount after 14 days is given as follows:
\(a_{14} = 20000(0.5)^{13}\)
\(a_{14} = 2.44\)
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first step for solving this quadratic equation?
x2+2x-11=4
Answers
Answer:
subtract 4 from both sides
Step-by-step explanation:
Given
x² + 2x - 11 = 4
First step : subtract 4 from both sides
x² + 2x - 15 = 0 ← in standard form
Second step : factorise the quadratic
(x + 5)(x - 3) = 0 ← in factored form
Third step : equate each factor to zero and solve for x
x + 5 = 0 ⇒ x = - 5
x - 3 = 0 ⇒ x = 3
Fourth step : state the solutions
solutions are x = - 5, x = 3
9x - 4 = 5x - 4 + 4x
Answers
Answer:
x = 1
Step-by-step explanation:
HOPE THIS HELPS
PLZZ MARK BRAINLIEST
Find the surface area of a cylinder with a height of 7 yd and a base diameter of 10 yd.
Use the value 3.14 for , and do not do any rounding.
Be sure
to include the correct unit.
7-yd
10 yd
Answers
if it has a base diameter of 10, that means its base radius is half that, or 5.
\(\textit{surface area of a cylinder}\\\\ SA=2\pi r(h+r)~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ h=7\\ r=5 \end{cases}\implies \begin{array}{llll} SA=2\pi (5)(7+5) \\\\\\ SA=10\pi (12)\implies SA=120\pi \\\\\\ \stackrel{using~\pi =3.14}{SA=376.8~yd^2} \end{array}\)
yoooooooooo help me i need the answer asap
A triangle has one side of length 4 and another of length 9. Which of the following are possible lengths of the third side?
3
4
5
6
10
11
13
14
15
Answers
Answer:
I think its 13 but im not possitive
Step-by-step explanation:
I would double check :}
Please help me and explain thank u
Answers
The solution to this expression (√5 - 3√2)(√5 + 3√2) is equal to -13.
What is a difference of two (2) squares?In Mathematics and Geometry, the standard form for a difference of two (2) squares is modeled or represented by this mathematical expression:
a² - b² = (a + b)(a - b).
Where:
a and b represent numerical values (numbers or numerals).
Based on the information provided, we have the following mathematical expression:
(√5 - 3√2)(√5 + 3√2);
By comparison, we have the following:
(√5 - 3√2)(√5 + 3√2) = (a + b)(a - b)
a² - b² = (√5)² - (3√2)²
a² - b² = 5 - 18
a² - b² = -13.
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if x=p+2 and y=p-3, show that x+y+1=2p and x-y-5=0
Answers
If \(x=p+2\) and \(y=p-3\), then
\(x+y = (p+2) + (p-3) = 2p - 1 \implies x+y+1=2p\)
and
\(x-y = (p+2)-(p-3) = 5 \implies x-y-5=0\)
as required.
Often, conditional probabilities are worded with what phrase?
"dependent"
"given that"
"either/or"
"mutually exclusive"
Answers
The correct phrase commonly used to word conditional probabilities is "given that." This phrase explicitly indicates the condition or event on which the probability calculation is based and emphasizes the dependence between events in the probability calculation.
Let's discuss each option in detail to understand why the correct phrase is "given that" when wording conditional probabilities.
"Dependent": The term "dependent" refers to the relationship between events, indicating that the occurrence of one event affects the probability of another event. While dependence is a characteristic of conditional probabilities, it is not the specific wording used to express the conditionality.
"Given that": This phrase explicitly states that the probability is being calculated based on a specific condition or event being true. It is commonly used to introduce the condition in conditional probabilities. For example, "What is the probability of event A given that event B has already occurred?" The phrase "given that" emphasizes that the probability of event A is being evaluated with the assumption that event B has already happened.
"Either/or": The phrase "either/or" generally refers to situations where only one of the two events can occur, but it does not convey the conditional nature of probabilities. It is often used to express mutually exclusive events, where the occurrence of one event excludes the possibility of the other. However, it does not provide the specific condition on which the probability calculation is based.
"Mutually exclusive": "Mutually exclusive" refers to events that cannot occur simultaneously. While mutually exclusive events are important in probability theory, they do not capture the conditionality aspect of conditional probabilities. The term implies that if one event happens, the other cannot occur, but it does not explicitly indicate the specific condition on which the probability calculation is based.
In summary, the correct phrase commonly used to word conditional probabilities is "given that." It effectively introduces the condition or event on which the probability calculation is based and highlights the dependency between events in the probability calculation.
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Your soccer team, Mill Valley, plays two games against Fairfield soccer team . The probability that your team wins the first game is 0.3. Of your team wins the first game, the probability that they also win the second game is 0.6. Of your team loses the first game, the probability that they win the second game is 0.3. Let the random variable X be the number of games won by your team, Mill Valley. Find the probability model for X.
Answers
The probability model for X is: X 0 1 2P(X) 0.49 0.33 0.18 .The probability model for X can be found by computing the probabilities associated with each value of X. X can take on the values 0, 1, or 2 because there are only two games.
Let's compute the probability that Mill Valley wins both games: P(X = 2) = P(win 1) × P(win 2 | win 1) = (0.3) × (0.6)
= 0.18
Let's compute the probability that Mill Valley wins one game. There are two ways to win one game: win the first game and lose the second, or lose the first game and win the second.
We can calculate these probabilities separately.
P(win 1) × P(lose 2 | win 1) = (0.3) × (0.4) = 0.12P(lose 1) × P(win 2 | lose 1)
= (0.7) × (0.3)
= 0.21
So, the probability of winning one game is P(X = 1) = 0.12 + 0.21
= 0.33
Probability of winning no games
Let's compute the probability that Mill Valley wins no games. This is the complement of the probability of winning at least one game.
So, P(X = 0) = 1 − P(X ≥ 1)
= 1 − P(X = 1) − P(X = 2)
= 1 − 0.33 − 0.18
= 0.49
Therefore, the probability model for X is: X 0 1 2P(X) 0.49 0.33 0.18
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The ratio 28:21 is equivalent to 4:x. What is the value of x?
Answers
Answer:
3
Step-by-step explanation:
28:21=
7(4):7(3)=
4:3
x=3
Hope this helps!
please help and simplify these expressions and show how u found them! i will mark brainliest !
Answers
Multiply the k and the 4 by 5
Now you have 5k+20-2k
Minus 2k from 5k
Now you have 3k+ 20 that’s the answer
2(3+d-1)
Multiply 3, d, and 1 by 2
Now you have 6+2d-2
Minus 2 from 6
Now you have 4+2d that’s the answe
Answer:
3) 3k+20
4) 4+2d
Step-by-step explanation:
You first have to distribute by multiplying the number next to the expression in the parenthesis. Then you combine like terms.
3) 5(k+4)-2k
Step 1: (5×k= 5k) (5×4=20) equation is now 5k+20-2k
Step 2: 5k-2k=3k since they both have the same variable you combine like terms. Equation is now 3k+20 (Final answer)
4) 2(3+d-1)
Step 1: (2×3=6) (2×d=2d) (2×[-1]=2) Equation is now 6+2d-1.
Step 2: (6-2=4) Since both 6 and 1 don't have any variable their like terms and you combine them. Equation is now 4+2d (Final answer)
Unit 7 polygons and quadrilaterals
Homework 7 trapezoids
** this is a 2-page document **
Directions: if each quadrilateral below is a trapezoid, find the missing measures
Answers
The value of ML in the given quadrilateral which is a trapezoid is 58 units.
What is a trapezoid?A polygon with only one set of parallel sides is called a trapezoid. The parallel bases of a trapezoid are another name for these parallel sides. Trapezoids have two additional sides that are not parallel and are referred to as their legs.
Trapezoids are defined differently by different people. A trapezoid can have one and only one pair of parallel sides, according to one school of mathematics, whereas another contends that a trapezoid can have several pairs of parallel sides. If we take into account the second definition, then a parallelogram is also a trapezoid under that definition.
We know that, A median on a trapezoid will be parallel to the bases, with a length equal to the sum of the bases divide by 2.
Thus,
45 = 3x + 11 + 10x - 12 / 2
45 = 13x - 1/ 2
90 = 13x - 1
91 = 13x
x = 7
Substitute the value of x in ML:
ML = 10x - 12
ML = 10(7) - 12
ML = 70 - 12
ML = 58
Hence, the value of ML in the given quadrilateral which is a trapezoid is 58 units.
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Drag the costs to the table to classify them as direct operational costs or costs
covered by business insurance.
Costs Covered by
Business Insurance
Direct Operational Costs
Answers
Rent, insurance, commissions, and other expenses could be added to the operating costs of a service-based business. Most of these expenses are regular daily expenses.
What are operating expenses?A continuing cost for maintaining a system, a business, or a product is known as an operating expense, operating expenditure, operational expenditure, operational expenditure, or OPEX.
The expense of creating or providing non-consumable pieces for the system or product is known as capital expenditure (capex).
For instance, the annual costs of paper, toner, power, and maintenance for a photocopier are opex, while the cost of the photocopier itself is capex.
Additional running costs for a service-based business could include rent, insurance, commissions, and other costs.
The majority of these costs are routine daily spending.
Costs that weren't directement necessary for those activities are referred to be non-operating expenses.
Therefore, rent, insurance, commissions, and other expenses could be added to the operating costs of a service-based business. Most of these expenses are regular daily expenses.
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Correct question:
A service business may have some additional operating expenses like rent, insurance, commissions, and so on. Most of these expenses are incurred on a ___ basis.
Through my homework, I was able to get the equations properly, however I am not being able to solve for the variables properly. \(2l + 2w = 292 \\ l - w = 36\)
Answers
substituting the value of L into equation (2) we have:
91-W=36
Therefore
W=91-36=55
determine the angle between 0 and 2π that is coterminal with 17pi/4
Answers
the angle between 0 and 2π that is coterminal with 17π/4 is π/4.
To find the angle between 0 and 2π that is coterminal with 17π/4, we need to find an equivalent angle within that range.
Coterminal angles are angles that have the same initial and terminal sides but differ by a multiple of 2π.
To determine the coterminal angle with 17π/4, we can subtract or add multiples of 2π until we obtain an angle within the range of 0 to 2π.
Starting with 17π/4, we can subtract 4π to bring it within the range:
17π/4 - 4π = π/4
The angle π/4 is between 0 and 2π and is coterminal with 17π/4.
what is equivalent'?
In mathematics, the term "equivalent" is used to describe two things that have the same value, meaning, or effect. When two mathematical expressions, equations, or statements are equivalent, it means that they are interchangeable and represent the same mathematical concept or relationship.
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Draw a A PQR if PQ = 6.5cm, m< PQR=75 °and m <PRQ=45° using ruler and compasses
only.
Answers
Answer:
Check your question well
Step-by-step explanation:
U didn't ask to draw angle p so how can we get the angle r
what is the product of (-2x+6)(3x-4)
Answers
Answer:
\(-6x^2+26x-24\)Step-by-step explanation:
\((-2x+6)(3x-4)=\)\((-2x)(3x)-4(-2x)+6(3x)-6(4)=\)\(-6x^2+8x+18x-24=\)\(-6x^2+26x-24\)state the domain and range of the following relation
x^2+y^2=16
Answers
The domain and range is [-4, 4] and [0, 4]
What is Domain and range?The domain of a function is the set of values that we are allowed to plug into our function.
The range of a function is the set of values that the function assumes.
x² + y² = 16
y = √16 - x²
For domain under root should not be negative quantity,
16 - x²≥0
16≥x²
So, x≤4 or x≥4
Thus, the domain is [-4, 4]
Range:
y is maximum at x=0, y=4
y is minimum at x=4, y=0
Thus, range = [0, 4]
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monica has a goal to complete a triathlon. the race consists of a 0.93-mile swim, a 24.8-mile bike ride, and a 6.2-mile run. in training, monica can swim at an average pace of 2.1mph. she rides her bike at an average of 18.8mph and runs at 5.1mph. estimate how long it will take her to complete the triathlon.
Answers
3.31 it will take her to complete the triathlon.
To find this, we can apply the equation distance = rate x time.
We are aware of the length and pace of each triathlon leg. Let's name them, respectively, d and r. We must locate the time, which we shall designate as t.
d = r x t or
t = d/r
The distance is 0.93, or 93/100, and the rate is 1.2, or 12/10 when swimming. As a result of our formula, we have:
12/10 divided by 93/100 yields the same result as
93/100 * 10/12
time to swim = 93/120 = 31/40
We have a distance of 24.8 miles, or 248/10, and a speed of 18.8 miles, or 188/10, for biking. Using the same formula, we obtain:
248/10 divided by 188/10 or
248/10 * 10/188 = 248/188 = 62/47 = time to bike
Finally, for the run, the distance is 6.2 = 62/10 and the rate is 5.1 or 51/10. Plugging this into the formula gives:
62/10 divided by 51/10 or
62/10 * 10/51 = 62/51 = time to run
= 31/40 + 62/47 + 62/51
=3.31
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Name the market angles in different ways
Answers
i hope my answers are ✅.......
Someone help I’m LAAAAZYYYY
Answers
Answer:
It is 8,7
Hope this helps!
Step-by-step explanation:
3 (1 + 5)² + 147 = 0
whats the answer?
Answers
Answer: 255
Step-by-step explanation:
equals to 255
The bases of the prism below are right triangles. If the prism's height measures 11 units and its volume is 130.9 units 3 3 , solve for x x.
Answers
The value of x is equal to 4.8 units.
How to calculate the volume of a triangular prism?In Mathematics and Geometry, the volume of a triangular prism can be determined or calculated by using the following formula:
Volume = base area × height of the prism.
Assuming the variable x represent the length of one side of its base. Since the height of this triangular prism measures 11 units and its volume measures 130.9 cubic units, the value of x can be determined as follows;
Base area of triangle = 1/2 × x × 5
Base area of triangle = 5x/2 square units.
Volume of triangular prism = base area × height of the prism.
130.9 = 5x/2 × 11
130.9/11 = 5x/2
11.9 = 5x/2
5x = 23.8
x = 4.8 units.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Write an expression that represents the height of a tree that beqins at 6.6 feet and increases by 2.2 feet per year. Let t represent the number of years
Answers
The tree grows a 2.2 feet per year and have an initial height 6.6. So the equation of the height will be 6.6+2.2t.
Let the increase in height be a linear equation,
Let H be the height of the tree, t represent number of years.
We do not know the coefficients.
So, let H = a + b t
When t=o, H = 6.6 , So a= 6.6
When t= 1 , H = 6.6 + 2.2t
So the equation for the height of the tree with respect to years will be
H = 6.6 + 2.2t
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solve the equation 56= 5v + 3v