Therefore, the first term of the sequence is 3.
What is sequence?A sequence is an ordered list of numbers, usually generated by a pattern or a rule. Each number in the sequence is called a term, and the position of a term in the sequence is called its index. Sequences can have a finite or infinite number of terms, and they can be defined in various ways, such as recursively (where each term is defined in terms of the previous terms) or explicitly (where the formula for each term is given).
Here,
The given sequence is an arithmetic sequence, where the common difference between consecutive terms is 4. To find the first term, we can use the formula for the nth term of an arithmetic sequence, which is:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the position of the term in the sequence, and d is the common difference.
For this sequence, we know that a6 = 23 (since 23 is the last term given in the sequence), n = 6, and d = 4. Substituting these values into the formula, we get:
a6 = a1 + (6 - 1)4
23 = a1 + 20
a1 = 23 - 20
a1 = 3
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Complete question:
What’s the first term for the given sequence: 3, 7, 11, 15, 19, 23?
a room has a wall with an r-value of 15 f sq.ft. hr/btu. the room is 13 feet long and 11 feet wide with walls that are 9 ft high. on a particular day the average outside temperature is 36 degrees f. how much heat is transferred through the walls on that day (in btus)? answer to two decimal places without a unit.
On that particular day, approximately 272.32 BTUs of heat were transferred through the wall.
To calculate the amount of heat transferred through the walls on a particular day, we can use the formula:
Q = U × A × deltaT
where Q is the heat transferred (in BTUs), U is the overall heat transfer coefficient (in BTU/hr·ft^2·°F), A is the area of the wall (in ft^2), and deltaT is the temperature difference between the inside and outside of the wall (in °F).
First, let's calculate the area of the wall. The room has four walls, but we are only interested in the heat transferred through the wall with an R-value of 15. We can calculate the area of this wall as follows:
Area = Length × Height = 13 ft × 9 ft = 117 ft^2
Next, we need to calculate the overall heat transfer coefficient, U. The U-value is the reciprocal of the R-value, so:
U = 1 / R = 1 / 15 = 0.0667 BTU/hr·ft^2·°F
Now we need to calculate the temperature difference between the inside and outside of the wall. We are given that the outside temperature is 36 °F, but we don't know the inside temperature. Let's assume that the inside temperature is 70 °F, which is a typical room temperature. Then the temperature difference is:
deltaT = 70 °F - 36 °F = 34 °F
Finally, we can plug these values into the formula:
Q = U × A × deltaT
Q = 0.0667 BTU/hr·ft^2·°F × 117 ft^2 × 34 °F
Q ≈ 272.32 BTUs
Therefore, on that particular day, approximately 272.32 BTUs of heat were transferred through the wall.
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in a distribution for which the mean is 30 and the standard deviation is 6, what percentage of all scores occur at 36 or above?
The percentage of all scores that occur at 36 or above is approximately 15.87% if the mean of the distribution is 20 and standard deviation is 6.
We can use the Z-score formula to standardize the value of 36 and find the corresponding area under the normal curve.
The Z-score for a score of 36 in this distribution is
Z = (X - μ) / σ
Z = (36 - 30) / 6
Z = 1
Using a standard normal table or calculator, we can find the area under the normal curve to the right of Z = 1
P(Z > 1) = 0.1587
This means that the percentage of all scores that occur at 36 or above is approximately 15.87%.
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help please its due in a few minuets
Answer:
use desmos graphing calculator it helps alot
Step-by-step explanation:
x^{2}-x-6=0 has two zeros/solutions
x^{2}-x-6=0 has 1 zero/solution
x^{2}-x-6=0 has no zero/solution
Is this true or false, explain how you know.
1/7 • 1/7 • 1/7 = 3/7
Answer: False
Step-by-step explanation:
1/7 x 1/7 = 1/49 x 1/7 = 1/343
a -foot-long footbridge has two diagonal supports that meet in the center of the bridge. each support makes a angle with a short vertical support. a diagram shows the footbridge, the diagonal supports and the vertical supports create two right triangles. at the top of the diagram a horizontal line is shown and labeled 20 feet. the two right triangles are shown below this horizontal line.the two longest legs of both right triangles touch at the center of the footbridge.the shortest leg of both right triangles represents the vertical support on each end of the footbridge. the hypotenuse of both right triangles form the diagonal supports that meet at the center of the footbridge. the full length of the footbridge is 20 feet. the vertical supports are not labeled. each diagonal support is labeled x. a 65-degree-angle is formed between the short vertical support and the diagonal support on each end of the bridge. what is the length x of a diagonal support, to the nearest tenth of a foot?
The length of a diagonal support (x) is approximately 18.4 feet to the nearest tenth of a foot.
To find the length x of a diagonal support, we will use the information given about the right triangles formed by the vertical supports, diagonal supports, and the horizontal line (representing half the length of the footbridge).
Since the full length of the footbridge is 20 feet, the horizontal line in each right triangle measures half of that, which is 10 feet.
The angle formed between the short vertical support and the diagonal support is given as 65 degrees.
We'll use the sine function to find the length x of the diagonal support.
In this case, sine(65) = opposite side (vertical support) / hypotenuse (diagonal support or x).
We first need to find the length of the vertical support.
To do this, we can use the tangent function.
In this case, tangent(65) = opposite side (vertical support) / adjacent side (10 feet).
Solving for the vertical support: vertical support = 10 * tangent(65) ≈ 17.1 feet.
Now, we can plug this value back into the sine equation: sine(65) = 17.1 / x.
Solving for x (the diagonal support): x = 17.1 / sine(65) ≈ 18.4 feet.
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Haywood Price purchased living room furniture for $1,860. He borrowed $1,200 and promised to repay the loan in 2 years. The finance charge was $308.25. To the nearest hundredth of a percent, what is the annual percentage rate?
The annual percentage rate is approximately 9.97% when rounded to the nearest hundredth of a percent.
How to Find the Annual Percentage Rate?To find the annual percentage rate (APR), we can use the following formula:
APR = (finance charge / loan amount) / (loan period in years) x 100%
First, let's calculate the loan amount by subtracting the finance charge from the total cost of the furniture:
Loan amount = $1,860 - $308.25 = $1,551.75
Next, let's convert the loan period of 2 years to the decimal form of 2:
Loan period = 2 years = 2.0 years
Now we can plug in the values into the formula and solve for the APR:
APR = ($308.25 / $1,551.75) / (2.0) x 100%
APR = 0.0997 x 100%
APR ≈ 9.97%
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assume the mapping t w p2 ! p2 defined by t a0 c a1t c a2t 2 d 2 a0 c .3 a1 c 4 a2/t c .5 a0 6 a2/t 2 is linear. find the matrix representation of t relative to the basis b d f1; t; t 2 g.
So, the matrix representation of T relative to the basis B = {f1, t, t²} is:
| 2.5 3 0 |
| 0 4 0 |
| 0 0 6 |
To find the matrix representation of the linear transformation T relative to the basis B = {f1, t, t²}, we need to apply T to each element of the basis and then express the result as a linear combination of the basis elements.
Let's apply T to each element of the basis B:
1. T(f1) = T(1) = 2(1) + 0.5(1) = 2.5
2. T(t) = T(c) = 3(c) + 4(c²) = 3f1 + 4t
3. T(t²) = T(c²) = 6(c²) = 6t²
Now, we can express the results as linear combinations of the basis elements:
1. T(f1) = 2.5f1 + 0t + 0t²
2. T(t) = 3f1 + 4t + 0t²
3. T(t²) = 0f1 + 0t + 6t²
Finally, we can form the matrix representation of T relative to the basis B by using the coefficients of the linear combinations as the columns of the matrix:
T matrix = | 2.5 3 0 |
| 0 4 0 |
| 0 0 6 |
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The matrix representation of t with respect to the basis b = [tex]{f1, t, t^2}[/tex] is:
[1/2 -4/3 1 ]
[1 8/3 0 ]
[0 -5/2 3/2 ].
To find the matrix representation of the linear transformation t with respect to the given basis, we need to apply t to each of the basis vectors and represent the result as a linear combination of the basis vectors.
The coefficients of these linear combinations will form the columns of the matrix representation.
First, we apply t to each of the basis vectors:
[tex]t(1) = a0 + a1t + a2t^2 = a0 + a1f1 + a2t - (a1/2)f1 - (3a2/2)t^2[/tex]
[tex]t(t) = a0 + (4/3)a1t + (5/6)a2/t = a0 + (4/3)t + (5/6)t^2 - (4/3)f1 - (5/2)t^2[/tex]
[tex]t(t^2) = a0 + (3/2)a2/t^3 = a0 + (3/2)t^2 - (3/2)f1[/tex]
Next, we represent these results as linear combinations of the basis vectors:
[tex]t(1) = (a0 - a1/2)f1 + (a1 + a2)t - (3a2/2)t^2[/tex]
[tex]= 1f1 + 0t + 0t^2 + (-1/2)f1 + 1t + 0t^2 + 0f1 - (3/2)t^2[/tex]
[tex]= (1/2)f1 + 1t - (3/2)t^2[/tex]
[tex]t(t) = (-4/3)f1 + (a0 + 4/3)t + (5/6)t^2[/tex]
[tex]= 0f1 + 1t + 0t^2 + (-4/3)f1 + (4/3)t + (5/6)t^2 + 0f1 + 0t - (5/2)t^2[/tex]
[tex]= (-4/3)f1 + (8/3)t - (5/2)t^2[/tex]
[tex]t(t^2) = f1 + (a0 - 3/2)t^2[/tex]
[tex]= 0f1 + 0t + 1t^2 + 1f1 + 0t + 0t^2 + 0f1 + (1/2)t^2[/tex]
[tex]= 1f1 + 0t + (3/2)t^2[/tex]
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The diameter of a circle is 10cm find the circumference to the nearest tenth
Answer c= Cm
Answer:
31.4 cm
Step-by-step explanation:
Circumference of circle = D x π
D = 10 cm
π = 3.14
Let's solve
10 x 3.14 = 31.4 cm
So, the circumference of the circle is 31.4 cm
Help me pls........................
All the possible angle measures are given as follows:
25º and 35º.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.On the first quadrant, we have that:
The cosine is greater than the sine for angles that are less than 45º.The cosine is equals to the sine for an angle of 45º.The cosine is less than the sine for angles that are greater than 45º.More can be learned about trigonometric ratios at brainly.com/question/24349828
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Please help me on this! I’ve been stuck on this since yesterday. I need help. :(
Answer:
x ≈ 48.6°
Step-by-step explanation:
using the sine ratio in the right triangle
sin x = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{21}{28}[/tex] = [tex]\frac{3}{4}[/tex] , then
x = [tex]sin^{-1}[/tex] ( [tex]\frac{3}{4}[/tex] ) ≈ 48.6° ( to the nearest tenth )
anova with replication is also known as? repeated measures between-group design mixed design one-way anova
this is "Repeated measures ANOVA, which is a statistical test used to analyze data from experiments in which the same participants are measured multiple times under different conditions.
What is a repeated measures ANOVA ?A repeated measures ANOVA is a statistical test used to analyze data from experiments in which the same participants are measured multiple times under different conditions. This design is sometimes referred to as a "within-subjects design" because each participant is measured within the same group.
Here are the steps for conducting a repeated measures ANOVA:
State the null and alternative hypotheses.Check the assumptions of the test, which include normality, sphericity, and homogeneity of variance.Calculate the within-subjects sum of squares (SSW), the between-subjects sum of squares (SSB), and the total sum of squares (SST).Calculate the degrees of freedom (df) for each of these sums of squares.Calculate the mean square (MS) for each source of variance by dividing the sum of squares by the degrees of freedom.Calculate the F ratio by dividing the between-subjects MS by the within-subjects MS.Determine the critical value for the F ratio using a table or software.Compare the calculated F ratio to the critical value. If the calculated F ratio is greater than the critical value, reject the null hypothesis and conclude that there is a significant effect of the independent variable on the dependent variable.If the null hypothesis is rejected, follow up with post hoc tests to determine which conditions differ significantly from each other.
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_____ show the change in one or more variables progressively across time. a. Histograms b. Pie charts c. Line graphs d. Bar charts e. Diagrams.
The correct answer is c. Line graphs show the change in one or more variables progressively across time.
Histograms show the distribution of data, pie charts show the proportions of a whole, bar charts compare categories or groups, and diagrams show the relationships between different elements.
The correct answer is c. Line graphs show the change in one or more variables progressively across time
A histogram is a visual depiction of data distribution in statistics. The histogram is shown as a collection of neighbouring rectangles, where each bar represents a certain type of data. Numerous fields use statistics, which is a branch of mathematics. Frequency, which can be expressed as a table and is known as a frequency distribution, is the repeating of numbers in statistical data.
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Line graphs show the change in one or more variables progressively across time. C
A line graph is a type of chart that displays data as a series of data points connected by straight lines.
The horizontal axis of the graph typically represents time, while the vertical axis represents the value of the variable being measured.
By connecting the data points with lines, a line graph can show the trend or pattern of change in the variable over time.
Line graphs are commonly used in fields such as economics, finance, and science to visualize trends in data over time.
They can be used to identify patterns, compare different variables, or forecast future trends.
In contrast, histograms, pie charts, bar charts, and diagrams are different types of charts that are used to display data in different ways.
Histograms are used to show the distribution of data within a single variable, while pie charts and bar charts are used to compare different categories or groups of data.
Diagrams are used to represent complex systems or relationships, often using visual symbols or icons to represent different components.
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some students checked 6 bags of doritos marked with a net weight of 28.3 grams. they carefully weighed the contents of each bag, recording the following weights (in grams): 29.2, 28.5, 28.7, 28.9, 29.1, 29.5. a) do these data satisfy the assumptions for inference? explain.
If some students checked 6 bags of Doritos marked with a "net-weight" of 28.3 grams, then these data satisfies all the three assumptions for inference (random condition, independent condition and normal condition.
To determine whether weight of bag satisfy the assumptions for inference, we need to consider the assumptions of statistical inference.
The assumptions are :
(i) Random Sampling: The data should be collected from a random sample, where each bag of Doritos has an equal chance of being selected.
This assumption is satisfied, because we assume that the "6-bags" are randomly selected.
(ii) Independence: The data points should be independent of each other, meaning that the weight of one bag of Doritos should not depend on the weight of another bag.
This assumption is satisfied as, bags were weighed separately and their weights were recorded without any influence from other bags.
(iii) Normality: The distribution of the data should be approximately normal.
This assumption is also satisfied , because histogram shows that there are "No-Outliers" and normal "quantile-plot" is linear.
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Juan can run 38 yards in 5 seconds. If he keeps the same pace, how many yards can he run in 30 seconds? Responses
Answer: 228 yards
Step-by-step explanation:
Answer:
228
Step-by-step explanation:
Find the volume of the rectangular prism.
9 yd
Pyd
4 yd
Answer:
To find the volume of a rectangular prism, we need to multiply its length, width, and height.
The given dimensions are:
Length = 9 yd
Width = Pyd (it's unclear what this means, so let's assume it's a typo and the intended value is 3 yd or 4 yd)
Height = 4 yd
Assuming the width is 3 yd, the volume of the rectangular prism is:
Volume = Length x Width x Height
Volume = 9 yd x 3 yd x 4 yd
Volume = 108 cubic yards
Assuming the width is 4 yd, the volume of the rectangular prism is
Volume = Length x Width x Height
Volume = 9 yd x 4 yd x 4 yd
Volume = 144 cubic yards
Therefore, the volume of the rectangular prism is either 108 cubic yards or 144 cubic yards, depending on the intended value of the width.
The expression (h) (b, + b₂) gives the area of a trapezoid, with b, and b, representing the two base lengths of a trapezoid and h representing the height. Find the area of a trapezoid with base lengths 4 in. and 6 in. and a height of 8 in. (Lesson 10.2)
Answer:
A = 40 in²
Step-by-step explanation:
the area (A) of a trapezoid is calculated as
A = [tex]\frac{1}{2}[/tex] h (b₁ + b₂)
where h is the height between the bases b₁ and b₂
here h = 8 , b₁ = 4 , b₂ = 6 , then
A = [tex]\frac{1}{2}[/tex] × 8 × (4 + 6)
= 4 × 10
= 40 in²
5 (-x-1)=x+21 - 6xn what is the solution to the following equation
You have a TV that is 30 inches tall and 40 inches wide. What is the length of the diagonal of the TV. (The measure from bottom left to top right of the TV)
2500 inches
1250 inches
50 inches
26.5 inches
Answer:
50 inches
Step-by-step explanation:
Using the Pythagorean theorem, we can find the length of the diagonal of the TV:
diagonal^2 = height^2 + width^2
diagonal^2 = 30^2 + 40^2
diagonal^2 = 900 + 1600
diagonal^2 = 2500
Taking the square root of both sides, we get:
diagonal = sqrt(2500)
diagonal = 50
Therefore, the length of the diagonal of the TV is 50 inches.
pls help with this question
Therefore, the original price of the suit was $950.
What is percent?Percent is a way of expressing a number as a fraction of 100, and is denoted by the symbol %. For example, 50% means 50 out of 100, or 0.5 as a decimal. Percentages are commonly used to express proportions, rates, discounts, and many other kinds of numerical information.
Here,
Let x be the original price of the suit. We know that the sale price, which is 34% less than the original price, is $627.
We can set up the equation:
x - 0.34x = 627
Simplifying this equation, we get:
0.66x = 627
Dividing both sides by 0.66, we get:
x = 950
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the major flaw of the linear probability model is that a. the actuals can only be 0 and 1, but the predicted are almost always different from that. b. the regression r2 cannot be used as a measure of fit. c. people do not always make clear-cut decisions. d. the predicted values can lie above 1 and below 0.
The major flaw of the linear probability model is, Option d, which allows predicted values to range from above 1 to below 0
The binary dependent variable, which accepts values of 0 and 1, and the independent variables are assumed to have a linear relationship under the linear probability model.
The anticipated values from the linear probability model, however, can range from 0 to 1, and in some circumstances, they can be either above or below 0. This goes against the dependent variable's probabilistic character and can produce inaccurate forecasts.
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what is 12 times the sum of a number v/u and 41
The expression 12 times the sum of the number v/u and 41 in the mathematical form will be 12(v/u + 41).
What is an expression?
The mathematical expression is the combination of numerical variables and operations denoted by addition, subtraction, multiplication, and division signs. Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also be used to denote the logical syntax's operation order and other properties.
Here, we have
Given: v/u and 41.
We have to find 12 times the sum of a number.
The sum of v/u and 41 is v/u + 41, then 12 times this sum is written as below:-
= 12(v/u + 41)
Hence, the expression 12 times the sum of the numbers v and 41 in the mathematical form will be 12(v + 41).
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You and your friend are standing back-to-back. Your friend runs 20 feet forward and then 15 feet right. At the same time, you run 12 feet forward and then 9 feet right. You stop and throw a baseball to your friend, who catches it. How far did you throw the baseball?
Answer:
40 ft
Step-by-step explanation:
the starting point is like the origin on the coordinate grid.
let's say forward for your friends is the positive y direction, and right for your friend is positive x direction.
the end point coordinates of your friend are then
(15, 20)
forward for you (facing in the other direction) is the the negative y direction. and going right for you is then actually going in the negative x direction.
your end point is therefore
(-9, -12)
the distance between both points is then per Pythagoras (and the derived distance formula) :
distance² = (x1 - x2)² + (y1 - y2)² =
= (15 - -9)² + (20 - -12)² = 24² + 32² =
= 576 + 1024 = 1600
distance = 40 ft
In Circle C, the diameter AB bisects the chord EF.
If m
In the circle the mesaure of ∠ACF is 7
What do you mean by Diameter ?The distance from the centre of a circle to any point on the boundary is called the radius. The radius is half of the diameter; 2r=d 2 r = d . The line segment that joins two points on the circle is a chord.
If diameter of a circle bisects a chord ,then it must be perpendicular to the chord.
∴ m∠ACF = 90°
(8x + 34)° = 90°
8x = 90 - 34
x = 56/8
x = 7
Hence,mesaure of ∠ACF = 7
Complete question : In Circle C, the diameter AB bisects the chord EF.
If m∠ACF = (8x+34)°, enter in the correct value of x.
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A homeowner bought a homeowner's insurance policy. The homeowner pays an annual $650 premium for property coverage with a deductible is $1,175 for each claim. A windstorm causes $42,500 in damages. If a claim for the full amount of damages is approved, how much will the homeowner's insurance company pay for the damages?
$42,500
$41,325
$41,975
$40,825
Answer:
The homeowner’s insurance company will pay for the damages after the homeowner pays the deductible of $1,175 for each claim. Therefore, the amount that the homeowner’s insurance company will pay for the damages is $42,500 - $1,175 = $41,325.
I hope that helps! (´▽`ʃ♡ƪ)
Step-by-step explanation:
Given vectors a=(1,2)b=(2,1) ,find 3a-3b
Find the measures of angle a and B. Round to the nearest degree.
The measure of angle A and angle B are 28.61 and 61.39 rounded to two decimal place, using trigonometric ratio of tangent for the respective angles.
What is trigonometric ratios?The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.
The basic trigonometric ratios includes;
sine, cosine and tangent.
tan B = 6/11 {opposite/adjacent}
B= tan⁻¹(6/11) {cross multiplication}
B = 28.6105.
tan A = 11/6 {opposite/adjacent}
A= tan⁻¹(11/6) {cross multiplication}
A = 61.3895.
Therefore, the measure of angles A and B are 28.61 and 61.39 rounded to two decimal place, using trigonometric ratio of tangent for the respective angles.
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The life, in years, of a certain type of electrical switch has an exponential distribution with an average life betaequals3. If 300 of these switches are installed in different systems, what is the probability that at most 70 fail during the first year?
The probability that at most 70 switches fail during the first year is low, at around 1.3%.
The problem involves the use of the exponential distribution to model the lifetime of a certain type of electrical switch. Specifically, we are given that the average life of these switches is beta = 3.
The exponential distribution is often used to model the time between occurrences of a certain event, such as the failure of a component. In this case, we can use it to model the time between failures of the electrical switches.
To answer the question, we need to find the probability that at most 70 switches fail during the first year. Let X be the number of switches that fail during the first year. We know that X follows a Poisson distribution with parameter lambda = (1/3)*300 = 100.
This is because the expected number of failures per year for each switch is 1/3 (since the average life is 3 years), and there are 300 switches installed.
Therefore, we can use the Poisson distribution to calculate the probability that at most 70 switches fail during the first year:
P(X <= 70) = e^(-100) * (100^0/0!) + e^(-100) * (100^1/1!) + ... + e^(-100) * (100^70/70!)
This can be calculated using a calculator or software such as Excel. The answer is approximately 0.013, or 1.3%. This suggests that the switches are fairly reliable, and that it is unlikely that a large number of them will fail in a short period of time.
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Can someone help me asap? It’s due today. Select all that apply.
I will give brainliest if it’s all correct!!
Answer:
3 & 4
Step-by-step explanation:
Since you have 3 choice, you want to pick the options that can represent 3 options.
1. False, since a coin only has 2 sides.
2. False, black and red is two halves of a deck, meaning you only have 2 options.
3. True, 6 is divisible by three, so it can be used. Assuming each choice is represented by two number.
4. True, 3 choices is evenly represented by 3 sections
The diagonal of a square is 10
inches long. What is the length, in inches, of each side of the square? Write the answer in simplified radical form.
7.071 is the length, in inches, of each side of the square.
What is meant by "square"?
The term "square" refers to a regular quadrilateral having four equal-length sides and angles. Angles in the square are at right angles or are separated by 90 degrees. The diagonals of the square are also equal and meet at a 90-degree angle.
Having equal length sides and right angles on all four, a square is a quadrilateral. Due to the fact that a square's sides are all the same length, it may be distinguished from other types of rectangles. Every square consequently becomes a rectangle since it is a quadrilateral with right angles at all four of its angles.
If the side od=f a square is s, the diagonal is s√2
This comes from Pythagoras as diagonal is
√s² + s² = √2s² = s√2
as such s√2 = 10
s = 10/√2
= 10 * √2/√2 * √2
= 5√2
5 * 1.4142 = 7.071
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Mr. Yara has 200 acres of land available to plant for three types of fruit trees (lemon, peach and
cherry). Based on the market survey, lemon has a selling price of $2 per kg; peach has a selling
price of $1. 5 per kg and cherry has a selling price of $4 per kg. The average yield per acre for
lemon, peach and cherry is 4, 6 and 2 tonne, respectively. The fertilizer requires per acre for
lemon, peach and cherry is 100, 100 and 50 kg, respectively. An acre of lemon and cherry
requires 10 labour hours each whereas an acre of peach requires 12 labour hours. Maximum
availability of labour is 20,000 hours. The cost of fertilizer is $2 per kg and the cost of labour is
$40 per hour. What is the optimal allocation of acre to the three types of fruit trees?
[1 tonne = 1000kg]
(a) Formulate a linear programming model to maximize the profit.
(b) Find the optimal solution by using simplex algorithm.
(a) The linear programming model to maximize the profit is written as,
Maximize Z = 2(4x1)(100) + 1.5(6x2)(100) + 4(2x3)(1000) - 2(100x1 + 100x2 + 50x3) - 40(10x1 + 12x2)
(b) The optimal solution is x1 = 0, x2 = 8/3, x3 = 8, x4 = 0, and the maximum profit is $40.
(a) Linear programming model:
Let x1, x2, and x3 be the number of acres allocated to lemon, peach, and cherry trees, respectively. Then, the objective function we want to maximize is:
Maximize Z = 2(4x1)(100) + 1.5(6x2)(100) + 4(2x3)(1000) - 2(100x1 + 100x2 + 50x3) - 40(10x1 + 12x2)
The first term represents the revenue from selling lemons, the second term represents the revenue from selling peaches, the third term represents the revenue from selling cherries, the fourth term represents the cost of fertilizer, and the last term represents the cost of labor.
The constraints are:
x1 + x2 + x3 ≤ 200 (total available land)
10x1 + 12x2 + 10x3 ≤ 20000 (total available labor hours)
x1, x2, x3 ≥ 0 (non-negativity constraints)
(b) Simplex algorithm:
Convert the linear programming model into standard form by introducing slack variables and expressing all variables in terms of these variables.
Maximize Z = 2(4x1)(100) + 1.5(6x2)(100) + 4(2x3)(1000) - 2(100x1 + 100x2 + 50x3) - 40(10x1 + 12x2)
Subject to:
x1 + x2 + x3 + x4 = 200
10x1 + 12x2 + 10x3 + x5 = 20000
x1, x2, x3, x4, x5 ≥ 0
In this case, the entering variable is x1, as it has the highest coefficient in the objective function.
Step 5 (continued):
1.6 0.1 0 1 -48 32 <- Z
0.4 0.9 1 1/10 0 40 <- x1
8.4 3.3 0 -1/10 1 1840 <- x5
The entering variable is x3, as it has the highest coefficient in the objective function.
The leaving variable is x1, as it has the smallest ratio of RHS to its coefficient in the x3 column.
x2 x1/12 x3 x5/120 x4 RHS
2.5 1/5 0 -1/60 0 40 <- Z
0.5 -1/5 1 1/60 0 8 <- x3
0.5 2/15 0 1/60 1/10 8/3 <- x2
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