Answer:
$ 922.25
Step-by-step explanation:
General equation for simple interest ... B(t)=B(0)[1+rt]B
(0)= inital amount r= simple annual interest rate t= number of yearsFor this problem,
B (2)=$850[0.0425×2}=850×1.085 =$922.25
after one year, a sample of radioactive material disintegrates to 90% of the original amount. for each part, round the answer to two decimal places. (a) find the half-life of the material. yr (b) find the time at which the material disintegrates to 80% of the original amount.
For a sample of radioactive material,
a) The half-life of the radioactive material with that it disintegrates to 90% of the original amountb is equals to the 6.58 years.
b) The time is 2.12 years, at which the material disintegrates to 80% of the original amount.
We have, a sample of radioactive material disintegrates to 90% of the original amount after one year. The term half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate. Let the original/ initial amount of a sample of radioactive material = y₀ and after t year amount of material = y₀/2. After one year, amount of sample of radioactive material = 90% of y₀. Now, radioactive decay equation form is y(t) = y₀( 0.90)ᵗ ,
We have to calculate value of t when y = y₀/2.
=> [tex]\frac{ y_0}{2} = y_0( 0.90)^t[/tex]
=> 1/2 = (0.90)ᵗ
Taking natural logarithm both sides
=> ln( 1/2) = ln ( 0.90)ᵗ
=> ln( 1/2) = t ln ( 0.90)
=> t = ln(1/2)/ln(0.90)
=> t = 6.579 ~ 6.58 years
b) Now, we have to determine value of t when material disintegrates to 80% of the original amount. Let t years be required time here. So, 80% of y₀ = y₀ ( 0.90)ᵗ
=> 0.80y₀ = y₀( 0.90)ᵗ
Taking natural logarithm both sides
=> ln( 0.80 ) = ln( 0.90)ᵗ
=> ln( 1/2) = t ln( 0.90)
=> t = ln(0.80)/ln(0.90)
=> t = 2.117 ~ 2.12years
Hence, required value of time is 2.12 years.
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earned a weekly wage of $1250. He paid 6% of this into a superannuation fund.
his weekly superannuation payment = $
Briefly explain what can you describe a term Derivative? Also
Give graphical explanation and real-world examples.
Answer:
Step-by-step explanation: Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is
(f(a + h)-f(a))/h
Now using the definition of derivative, we can write
f
'
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
which is also the instantaneous rate of change of the function f(x) at a.
Now, for a very small value of h, we can write
f'(a) ≈ (f(a+h) − f(a))/h
or
f(a+h) ≈ f(a) + f'(a)h
This means, if we want to find the small change in a function, we just have to find the derivative of the function at the given point, and using the given equation we can calculate the change. Hence the derivative gives the instantaneous rate of change of a function within the given limits and can be used to find the estimated change in the function f(x) for the small change in the other variable(x).
Approximation Value
Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is
L(x) = f(a) + f'(a)(x−a)
The tangent will be a very good approximation to the function's graph and will give the closest value of the function. Let us understand this with an example, we can estimate the value of √9.1 using the linear approximation. Here we have the function: f(x) = y = √x. We will find the value of √9 and using linear approximation, we will find the value of √9.1.
We have f(x) = √x, then f'(x) = 1/(2√x)
Putting a = 9 in L(x) = f(a) + f'(a)(x−a), we get,
L(x) = f(9) + f'(9)(9.1−9)
L(x) = 3 + (1/6)0.1
L(x) ≈ 3.0167.
This value is very close to the actual value of √(9.1)
Hence by using derivatives, we can find the linear approximation of function to get the value near to the function.
2. She proceeds to sell the books for R80.00 each. Complete the input/output table below: 2.1 2.3 2.4 Number of Books Fixed Costs Variable Costs Total Costs Printer's Costs 2.4.1 2.4.2 0 A 0 C 0 50 950 902.50 100 1400 950 B 1852.50 2755 D 150 950 2707.50 3657.50 4200 200 950 3 610 4 560 2.2 Complete the graphs for the cost of using a printer and the Total Costs on ANNEXURE A. 5600 Indicate the Point where the graphs meet with the letter A. This point is called the break even point. Use your graph to indicate the following values at this point: The cost of the books. Number of books. (4) (
Answer:
I'm sorry, but the information you provided is not clear enough for me to create the required input/output table and graphs on ANNEXURE A. Could you please provide more context and details on the problem? What are the fixed costs, variable costs, and printer's costs for? Are there any other costs involved in selling the books? How many books are initially available for sale? Without this information, I won't be able to provide an accurate solution to your problem.
Step-by-step explanation:
Evaluate the integral by making an appropriate change of variables.
[tex]\int\ \int\R {12sin(16x^2+64y^2)} \, dA[/tex]
where R is the region in the first quadrant bounded by the ellipse 16x^2 + 64y^2 = 1
Answer:
To evaluate the integral over the region R, we can use the change of variables:
u = 4x
v = 2y
This gives us:
x = u/4
y = v/2
The Jacobian of this transformation is:
| ∂x/∂u ∂x/∂v | = | 1/4 0 |
| ∂y/∂u ∂y/∂v | | 0 1/2 |
So the Jacobian determinant is |J| = (1/4)(1/2) = 1/8.
Using this transformation, the region R is mapped onto the unit circle in the uv-plane, and the equation of the ellipse becomes:
u^2 + v^2/4 = 1/16
The integral becomes:
∫∫R 4x^2 e^(4xy) dA
= 2∫∫S u^2 e^uv/2 (1/8) dA
= (1/4) ∫∫S u^2 e^v/2 dA
where S is the unit circle in the uv-plane.
Now we can use polar coordinates in the uv-plane, with u = r cosθ and v = r sinθ. The integral becomes:
(1/4) ∫∫S r^2 cos^2θ e^(r sinθ/2) r dr dθ
= (1/4) ∫0^2π ∫0^1 r^3 cos^2θ e^(r sinθ/2) dr dθ
The inner integral can be evaluated by integration by parts, letting u = r^2 cos^2θ and dv = e^(r sinθ/2) r dr. This gives:
∫ r^3 cos^2θ e^(r sinθ/2) dr
= r^3 cos^2θ (-2/θ) e^(r sinθ/2) + 2/θ ∫ r^2 cos^2θ e^(r sinθ/2) dr
The integral on the right-hand side can be evaluated by another integration by parts, letting u = r^2 cos^2θ and dv = e^(r sinθ/2) dr, which gives:
∫ r^2 cos^2θ e^(r sinθ/2) dr
= r^2 cos^2θ (-2/θ) e^(r sinθ/2) + 4/θ^2 ∫ r cos^2θ e^(r sinθ/2) dr
We can substitute these results back into the original integral and simplify to get:
∫∫R 4x^2 e^(4xy) dA
= (1/4) ∫0^2π ∫0^1 r^3 cos^2θ e^(r sinθ/2) dr dθ
= (1/2π) ∫0^π ∫0^1 r^3 cos^2θ e^(r sinθ/2) dr dθ
Now we can evaluate the inner integral:
∫0^1 r^3 cos^2θ e^(r sinθ/2) dr = (1/2) ∫0^1 r^2 e^(r sinθ/2) d(r^2)
= (1/2) ∫0^1 u^(1/2) e^(u sinθ/2) du
Letting t = u sin(θ/2) and using the identity sin(θ/2) = 2
A triangle has side lengths of ( 1.3 t + 7.9 u ) (1.3t+7.9u) centimeters, ( 2.4 t − 4.8 v ) (2.4t−4.8v) centimeters, and ( 6.8 v − 9.8 u ) (6.8v−9.8u) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
Step-by-step explanation:
why did you mention the definition of every side twice ?
I think we need only one per side right ?
because otherwise we would have to do all 3 multiplications before summing things up.
I sorted the terms for the sum based on their variable.
after all, the perimeter of a triangle is the sum of all 3 sides.
1.3t + 7.9u
2.4t - 4.8v
- 9.8u +6.8v
--------------------------
3.7t - 1.9u + 2v cm
0.4(2x+ 1/2 )=3[0.2x+(−2)]−4
Use this regular pentagon to answer the questions.
A regular pentagon with center point A is shown. Line are drawn from each point to center point A. Triangle A B C is formed.
It displays a regular pentagon with centre point A. From each point, a line is traced to the centre point A. Triangles ABC are made. One of the regular pentagon's central angles measures 72°. In triangle ABC, one of the congruent base angles has a measure of 54.
In a regular pentagon, each of the five central angles has the same measure, which can be found using the formula:
[tex]Central angle = 360° / Number of sides[/tex]
Since we have a regular pentagon, the number of sides is 5, so:
[tex]Central angle = 360[/tex]°/5
central angle = 72°
Therefore, each central angle in the regular pentagon measures 72°.
Now let's look at triangle ABC. Since the pentagon is regular, all the line segments from the vertices to the centre are of equal length, so triangle ABC is an isosceles triangle with base angles that are congruent. Give one of these basic angles a measure of x.
Since the sum of the interior angles in any triangle is 180°, we can write:
x + x + angle at vertex = 180°
But we know that the angle at the vertex is a central angle of the pentagon, and so it measures 72°. Thus, we can substitute 72° for the angle at the vertex, and we have:
2x + 72° = 180°
Subtracting 72° from both sides, we get:2x = 108°
Dividing by 2, we find:
x = 54°
Therefore, each of the congruent base angles in triangle ABC measures 54°.
The complete question is:-
Use this regular pentagon to answer the questions. A regular pentagon with center point A is shown. Line are drawn from each point to center point A. Triangle A B C is formed. What is the measure of one of the central angles in the regular pentagon? ° What is the measure of one of the congruent base angles in triangle ABC? °
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Jordyn’s big sister gives her a 10-yard head start before they start racing. If Jordyn can run 1 yard per second, make a table of values, graph, and an equation that represents how far Jordyn can go.
A. (-8) times 2/3 B. (-8) times (-2/3) C. 9/4 divided by -3/4 D. 1/3= -5 What is A?
Answer: -16/3
Step-by-step explanation:
-8 * 2/3
-8 * 2 = -16
Thus, A = -16/3
what is the answer to this
Porter's points per basketball game are normally distributed with a standard deviation of 6 points If Porter scores 50 points, and the z-score of this value is 4, then what is his mean points in a game? Do not include the units in your answer. For example, if you found that the mean is 33 points, you would enter 33. Provide vour answer below:
The mean points per game for Porter is 26 points.
To find the mean, we use the formula for the z-score:
z = (x - mu) / sigma
where z is the z-score, x is the observed value, mu is the mean, and sigma is the standard deviation. Rearranging this formula, we get:
mu = x - z * sigma
Plugging in the values given in the problem, we have:
mu = 50 - 4 * 6 = 26
This would be the mean if the z-score is 4.
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Helppp
Use trigonometry to find EF to the nearest tenth.D = 36 , DE = 16
Considering the figure, the length of EF is solved to be 22.0
How to find EFThe length EF of the right triangle is solved using trigonometry as follows
Considering the figure and the giving sides we use the trigonometric tangent by using the formula
tan (angle D) = DE / EF
plugging in the values
tan 36 = 16 / EF
EF = 16 / tan 36
EF = 22.022
EF = 22.0 (to the nearest tenth)
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45. Graph the line that passes through the points (-2, 5) and
(5,-2). What are the x- and y-intercepts for this line?
Answer:
Step-by-step explanation:
Can I get help with this question?
Answer:
6mn12mnStep-by-step explanation:
You want to identify factors of 12mn that appear on the list.
FactorsOften, we are interested in prime factors of a number or expression. These are integers or expressions that are divisible only by themselves and 1. Here, the expression 12mn written in terms of its prime factors is ...
2 · 2 · 3 · m · n
The product of any subset of these factors is a factor of 12mn. Such products include ...
2·3·m·n = 6mn2·2·3 = 12m·n = mnEn un teatro, 10 entradas de adulto y 9 de niños cuestan $81.500; 17 entradas de niños y 14 de adultos cuestan $134.500. Hallar el precio de una entrada de adulto y una de niño.
Ayuda pls es para mañana
The price of one adult ticket is approximately $3,977.27, and the price of one child ticket is approximately $4,636.37.
How to solveLet's use the system of linear equations to solve this problem.
Let x be the price of an adult ticket and y be the price of a child ticket. We can create two equations based on the information given:
10x + 9y = 81,50014x + 17y = 134,500To solve this system of linear equations, we can use either substitution or elimination. In this case, we'll use elimination.
First, multiply the first equation by 17 and the second equation by 9 to make the coefficients of y the same:
170x + 153y = 1,385,500126x + 153y = 1,210,500Now, subtract the second equation from the first equation:
(170x + 153y) - (126x + 153y) = 1,385,500 - 1,210,500
44x = 175,000
Now, divide both sides of the equation by 44 to solve for x:
x = 175,000 / 44
x ≈ 3,977.27
Now that we have the price of an adult ticket, we can plug the value of x back into either equation to solve for y.
We'll use the first equation:
10(3,977.27) + 9y = 81,500
39,772.70 + 9y = 81,500
Now, subtract 39,772.70 from both sides of the equation:
9y = 41,727.30
Now, divide both sides of the equation by 9 to solve for y:
y = 41,727.30 / 9
y ≈ 4,636.37
So, the price of one adult ticket is approximately $3,977.27, and the price of one child ticket is approximately $4,636.37.
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The question in English is:
In a theater, 10 adult tickets and 9 children's tickets cost $81,500; 17 tickets for children and 14 for adults cost $134,500. Find the price of one adult and one child ticket.
Where do the medians of the triangle intersect?
Answer:
The medians intersect at 4-4 (i think).
Step-by-step explanation:
Every triangle has three medians and they all intersect in the triangles centroid. i believe that the medians intersect at 4-4.
i am not a very trustworthy source so you should probably ignore this awnser.
A car is originally worth $43,500. It takes 12 years for this car to totally depreciate a. Write a straight line depreciation equation that models the situation. b.how long will it take for this car to be worth 25% of its value? C.how much will the car be worth in 10 years?
Answer:
A: The straight-line depreciation equation is a linear equation of the form:
Value = initial value - (rate of depreciation) x (age)
Where the "rate of depreciation" is the amount by which the value decreases each year. In this case, the initial value is $43,500, and the car depreciates over a period of 12 years. So, you can calculate the rate of depreciation as:
Rate of depreciation = (initial value - final value) / (age)
Where the final value is zero (since the car is totally depreciated after 12 years). Therefore, you have:
Rate of depreciation = ($43,500 - $0) / 12 years = $3,625 per year
Substituting this into the formula, you get:
Value = $43,500 - $3,625 x (age)
Where "value" is the current value of the car after "age" years.
B: You want to find how long it will take for the car to be worth 25% of its value. Let's call this time "t". Then you have:
0.25($43,500) = $10,875 = $43,500 - $3,625t
Solving for "t", you get:
$3,625t = $43,500 - $10,875 = $32,625
t = $32,625 / $3,625 = 9 years
Therefore, it will take 9 years for the car to be worth 25% of its value.
C: You want to find how much the car will be worth in 10 years. Substituting "age = 10" into the equation derived in part A, you get:
Value = $43,500 - $3,625 x 10 = $7,750
Therefore, the car will be worth $7,750 in 10 years.
find the area of the sector
(70 points)
will mark brainly
Answer:
the area of the sector in grey is 100.48 square feet.
Step-by-step explanation:
Area of a circle = radius × radius × π
(π ≌ 3.14)
So
the area of the circle in the picture = 16 × 16 × π = 803.84
As the angle in the grey sector is 45°, so the proportion of the grey sector in the circle is
45°/360° = 1/8
Hence why
the area of the grey sector is 1/8 of the area of the circle, which is
803.84 × 1/8 = 100.48
A bag holds 13 marbles. 6 are blue, 2 are green, and 5 are red.
Match the events on the left with probabilities on the right
a) 8/13 of the marbles are blue or green, 61/5%
b) Blue and green replacing all the marbles would be 7.7%
c) 6/13 of the marbles are blue, 46/2%
d) (2/13)*(6/13) would be a chance of getting a green marble plus another without replacing, 7.1%
Mr Kramer buys a new car for 24,000 dollars. He will pay a sales tax of 0.06 dollars for every dollar of the price. He will pay a property tax of 14.62 dollars for every 1000 dollars. What are the total taxes on his new car?
The total taxes on Mr. Kramer's new car are:
1,440 + 351.48 = 1,791.48 dollars.
What is the rate?
A rate is a measurement of how one quantity changes with respect to another quantity, usually expressed as a ratio. For example, if a car travels 120 miles in 2 hours, its speed (or rate) is 60 miles per hour (120 miles divided by 2 hours).
The sales tax rate is 0.06, which means Mr. Kramer will pay 0.06 dollars for every dollar of the price.
The price of the car is 24,000 dollars, so the sales tax will be:
0.06 * 24,000 = 1,440 dollars
The property tax rate is 14.62 dollars for every 1000 dollars of the price.
To calculate the property tax on the car, we need to divide the price by 1000 and then multiply by the tax rate:
(24,000 / 1000) * 14.62 = 351.48 dollars
Therefore, the total taxes on Mr. Kramer's new car are:
1,440 + 351.48 = 1,791.48 dollars.
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4. Explain how you would create and use a spinner to simulate the probability of an event
with the following probabilities:
Percent chance: 20% green, 1/4 blue,2/5 yellow, and 15% orange
To get the given probabilities of the event we can use a spinner with 20 green, 25 blue, 40 yellow, and 15 orange parts.
What is probability?Probability is a way to gauge how likely or unlikely something is to happen. A number between 0 and 1, where 0 denotes an improbable event and 1 denotes a certain event, is used to convey it. The likelihood of an occurrence may be determined by dividing the positive outcomes by the entire number of possible outcomes. For instance, the likelihood of receiving heads on a fair coin flip is 50% since there is only one positive event (heads) out of two possible possibilities (heads or tails). The general probability calculation formula is:
Number of likely outcomes divided by the total number of possible outcomes is how you calculate an event's probability.
To get the given probabilities of the event we can use a spinner with 20 green, 25 blue, 40 yellow, and 15 orange parts.
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What is the difference between the longest pencil and the shortest pencil >:(
Triangle ABC is shown. Use the graph to answer the question.
triangle ABC on a coordinate plane with vertices at negative 8 comma 1, 0 comma 1, negative 4 comma 5
Determine the coordinates of the image if triangle ABC is translated 7 units to the right.
A′(−13, 1), B′(−7, 1), C′(−11, 5)
A′(−6, −6), B′(0, −6), C′(−4, −2)
A′(−6, 8), B′(0, 8), C′(−4, 12)
A′(−1, 1), B′(7, 1), C′(3, 5)
The coordinates of the image after the translation are:
A′(−1, 1), B′(7, 1), C′(3, 5)
So the answer is (D) A′(−1, 1), B′(7, 1), C′(3, 5).
What are coordinates ?
Coordinates are pairs of numbers that locate a point in a space or on a plane. In two-dimensional space, or the Cartesian plane, a point is located by an ordered pair of real numbers (x, y), where x represents the horizontal distance and y represents the vertical distance. These numbers are called the coordinates of the point.
The horizontal line is called the x-axis, and the vertical line is called the y-axis. The point where the x-axis and y-axis intersect is called the origin, and it is assigned the coordinates (0, 0).
According to the question:
To translate a figure, we add or subtract a constant value to the x and/or y coordinates of each vertex. In this case, we want to translate triangle ABC 7 units to the right. So, we will add 7 to the x-coordinate of each vertex.
The coordinates of A are (-8, 1). Adding 7 to the x-coordinate gives:
A′ = (-8 + 7, 1) = (-1, 1)
The coordinates of B are (0, 1). Adding 7 to the x-coordinate gives:
B′ = (0 + 7, 1) = (7, 1)
The coordinates of C are (-4, 5). Adding 7 to the x-coordinate gives:
C′ = (-4 + 7, 5) = (3, 5)
Therefore, the coordinates of the image after the translation are:
A′(−1, 1), B′(7, 1), C′(3, 5)
So the answer is (D) A′(−1, 1), B′(7, 1), C′(3, 5).
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Which of the following is the same as 25% of 84?
a 84÷5
b) 84÷4
c) 84÷3
d) 84÷2
Answer:
(b)
Step-by-step explanation:
25 percent of 84 equals 21
84 divided by 4 equals 21
Answer:
84 divided by 4
Step-by-step explanation:
25% of 84 is just (0.25)(84) which is 21
84 divided by 4 is also 21
PLEASE HELP FAST 100 POINTS PLUS WILL GIVE BRAINLIEST
Which of the following tables represents a linear function?
x 2 2 2 2 2
y −3 −2 −1 0 1
x −2 −1 0 2 4
y −4 −2 −1 0 1
x −6 −2 0 1 3
y −7 negative one third −5 negative four thirds 1
x −3 −1 1 3 5
y −7 negative nine halves −2 one half 3
The 3rd one
Step-by-step explanation:
I'm 99.9% sure it's right
I'm really sorry if not!
hope this helps
Table 3 represents a linear function.
How to determine the table that represents a linear function?
In a linear function, the change in the y-value is proportional to the change in the x-value by a constant rate, known as the slope. We can calculate the slope of Table 3 by selecting any two points and using the formula:
slope = (y2 - y1) / (x2 - x1)
For example, if we choose the points (-2, -4) and (2, 0), we get:
slope = (0 - (-4)) / (2 - (-2)) = 4 / 4 = 1
This means that for every increase of 1 in the x-value, the y-value increases by 1. We can confirm that this holds for all the other points in Table 3 as well, indicating that it represents a linear function.
The other tables do not represent linear functions because the change in the y-value is not proportional to the change in the x-value at a constant rate.
Production of a 5,350 linear-foot irrigation system for Thompson Orchards (job no. 399) was started and completed in March of the current year.
The production of a 5,350 linear-foot irrigation system for Thompson Orchards, job no. 399, was started and completed in March of the current year.
This means that the irrigation system was manufactured and delivered to the customer within that month. It is important to note that completing the production does not necessarily mean that the job is entirely finished, as there may be additional steps involved such as installation and testing. The completion of this job is a significant milestone for Thompson Orchards as it allows for efficient watering of their crops, which is crucial for their growth and productivity. The irrigation system was likely designed and manufactured to meet specific requirements, taking into consideration factors such as the type of crops grown, soil type, and climate. For the manufacturer, completing this job on time and to the customer's satisfaction is a testament to their expertise and ability to deliver high-quality products. This successful project can lead to repeat business and positive word-of-mouth recommendations. Overall, the completion of the irrigation system for Thompson Orchards is a significant accomplishment for both the manufacturer and the customer.
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In circle J, mLK 111°. Solve for æ if m/LJK= (10x38)°. If necessary, round your answer to the nearest tenth.
Given the information about the circle, the value of x is 7.3
What is the easy definition of a circle?A circle is simply a round shape that has no corners or line segments. It is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
The value of x based on the information given will be:
10x + 38 = 111
10x = 111 - 38
10x = 73
Divide
x = 73 / 10
x = 7.3
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Bethany can mow her family's lawn in 4 hours. Her brother Colin can mow the lawn in 3 hours. Which equation can
be used to find the number of hours, x, it would take for Bethany and Colin to mow the lawn together?
Bethany
Sess
Colin
3+4=X
x-x-1
0-3-1
Mark this and return
Rate
(Lawns per Hour)
1
4
13
Time
(Hours)
X
Save and Exit
Fraction Completed
Next
4X
1
3X
Submit
After answering the presented question, we can conclude that this equation depicts Bethany and Colin's combined work rate, where 1/4 represents Bethany's work rate.
What is equation?In mathematics, an equation is a statement that states the equality of two expressions. An equation consists of two sides separated by an algebraic equation (=). The argument "2x + 3 = 9," for example, states that the sentence "2x Plus 3" equals the value "9." The goal of solving equations is to find the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complex, linear or nonlinear, and contain one or more parts. In the equation "x² + 2x - 3 = 0," the variable x is raised to the second power. Lines are used in many areas of mathematics, including algebra, calculus, and geometry.
The following equation can be used to calculate the number of hours, x, required for Bethany and Colin to mow the lawn together:
1/4 + 1/3 = 1/x
This equation depicts Bethany and Colin's combined work rate, where 1/4 represents Bethany's work rate (in lawns per hour) and 1/3 represents Colin's work rate (in lawns per hour). When they mow the grass together, the equation makes the total of their individual work rates equal to their combined work rate. Solving for x gives us the number of hours it would take them to finish the job if they worked together.
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CAN SOMEONE HELP WITH THIS QUESTION?
The box with the most volume has 20 centimetres of side length and 20 centimetres of height. The box has a volume of: 8000 cubic centimetres.
We need to employ optimization strategies to determine the box with the highest feasible volume, which has a square base and an open top. Call the box's height h and the square base's side length x, respectively.
V = [tex]x^(2h)[/tex] gives the volume of the box.
The box's overall surface area will be 1600 square centimetres since the material used to create it has a surface area of 1600 square centimetres.
The box's surface area is made up of the areas of its four sides and base (x2) (4xh). We thus have:
[tex]x^2[/tex]+ 4xh = 1600
The volume of the box, which is given by V = [tex]x^(2h)[/tex], is what we wish to maximise.
We can determine h in terms of x using the equation above:
h = (1600 - [tex]x^2[/tex]) / (4x) (4x)
This result is obtained by replacing h with this equation in the volume formula:
V = [tex]x^2[/tex](1600 - [tex]x^2[/tex]) / (4x) (4x)
If we simplify this expression, we get:
V = 400x - [tex]0.25x^3[/tex]
Now, in order to determine the crucial places, we can take the derivative of V with respect to x and put it equal to zero:
[tex]dV/dx = 400 - 0.75x^2 = 0[/tex]
As a result of solving this equation for x,
x = 20
The result of replacing h in the equation with x = 20 is:
h = (1600 - 400) / (4 * 20) = 20
As a result, the box with the maximum volume has 20 centimetres of height and 20 centimetres of side length. The box has a volume of:
8000 cubic centimetres are equal to V = 202 * 20.
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