A farmer has 1,416 feet of fencing available to enclose a rectangle area bordering a river. No fencing is required along the river. Let x represent the length of the side of the rectangular enclosure that is perpendicular A(x)= Find the dimensions that will maximize the area. The length of the side rectangle perpendicular to the river is and the length of the side of the rectangle parallel to the river is.What is the maximum area?
Answers
The dimensions that will maximize the area are x = 354 ft and y = 708 ft
The length of the side rectangle perpendicular to the river is 354 ft
The length of the side of the rectangle parallel to the river is 708 ft
The maximum area = 250632 ft²
Explanation:Given:
The length of the fencing = 1416 ft
The length of the side rectangle perpendicular to the river = x
To find:
The dimensions that will maximize the area
To determine the dimensions, we will make an illustration of the given information:
let the length o the rectangle parallel to the river = y
Length of the for the enclosed area = Perimeter of the enclosed area
Perimeter of the enclosed area = x + x = y = 2x + y
[tex]1416=2x+y\text{ . . .\lparen1\rparen}[/tex]Area of the rectangle = length × width
length = y, width = x
let the Area of the rectangle = A(x)
[tex]A(x)\text{ = xy . . . \lparen2\rparen}[/tex]To get the expression for A(x), we will make y the subject of the formula in equation (1):
y = 1416 - 2x
substitute for y in equation (2):
[tex]\begin{gathered} A(x)\text{ = x\lparen1416 - 2x\rparen} \\ \\ A(x)\text{ = 14166x - 2x}^2 \end{gathered}[/tex]To get the maximum dimension, we will differentiate with respect to x:
[tex]\begin{gathered} A^{\prime}(x)\text{ = 1416 - 4x} \\ \\ At\text{ maximum, A'\lparen x\rparen = 0:} \\ 1416\text{ - 4x = 0} \\ 1416\text{ = 4x} \\ x\text{ = }\frac{1416}{4} \\ x\text{ = 354} \end{gathered}[/tex]substitute for x in equation (1):
[tex]\begin{gathered} 1416\text{ = 2\lparen354\rparen + y} \\ 1416\text{ - 708 = y} \\ y\text{ = 708} \end{gathered}[/tex]The dimensions that will maximize the area are x = 354 ft and y = 708 ft
The length of the side rectangle perpendicular to the river is 354 ft
The length of the side of the rectangle parallel to the river is 708 ft
The maximum area = 354 × 708
The maximum area = 250632 ft²
Related Questions
what is slope of the line parallel to the given line(I know the steps no explanation just need work:)
Answers
Given the equation of the line :
[tex]4x-3y=-12[/tex]To find the slope, solve the equation for y:
[tex]\begin{gathered} 4x-3y=-12 \\ -3y=-4x-12 \end{gathered}[/tex]Divide both sides by -3 :
[tex]\begin{gathered} \frac{-3y}{-3}=\frac{-4x}{-3}-\frac{12}{-3} \\ \\ y=\frac{4}{3}x+4 \end{gathered}[/tex]So, the slope of the given line = 4/3
And the slope of the line parallel to the given line = 4/3
Because the parallel line have the same slopes
So, the answer is : slope = 4/3
R's sleep log: 8.5 8 9.5 9 7.5 8.5B's sleep log 6.5 7 7.5 6.5 12 7.5which statement is NOT true about the data provided?a. both sets of data have an interquartile range of 1b. both measures of center for R's data have a value of 8.5c. the shape of both data distributions are non-symmetricd. the IQR of B's data is best used to describe the spread because of the outlierR's sleep log: 8.5 8 9.5 9 7.5 8.5B's sleep log 6.5 7 7.5 6.5 12 7.5
Answers
Answer
C. The shape of both data distributions are non-symmetric
Step-by-step explanation
Ordering both data sets from least to greatest, we get:
R's sleep log:7.5 8 8.58.5 99.5
B's sleep log: 6.5 6.577.57.512
We can see that B's data is concentrated at the lower values (12, the maximum, is an outlier). Then, the shape of B's data distribution is non-symmetric. But, in R's data every value is equidistant from the other ones, in consequence, the shape of R's data distribution is symmetric.
A telemarketers computer selects phone numbers at random. The telemarketer has recorded the number of respondents in each age bracket for one evening in the following table. What is the probability that the next respondent will be between 18 and 25? Enter a fraction or round your answer to four decimal places, if necessary.Under-18 =1818-25=2526-35=4636-45=56Over 45=54
Answers
The probability P of an event e can be found using the formula:
[tex]P(an\text{ event e occurring\rparen= }\frac{Number\text{ of occurrence of an event}}{Total\text{ number of occurrence}}[/tex]Required: The probability that the next respondent will be between 18 and 25
Number of respondents between the ages of 18 and 25 = 25
Total number of respondents :
[tex]\begin{gathered} =\text{ 18 + 25 + 46 + 56 + 54} \\ =\text{ 199} \end{gathered}[/tex]Hence, the probability of a selecting a respondents in the age range 18 to 25 is:
[tex]\begin{gathered} P\text{ = }\frac{25}{199} \\ =\text{ 0.125628} \\ \approx\text{ 0.1256} \end{gathered}[/tex]Answer: 0.1256
the table indicates the table indicates a man use data usage over the last 4 months positive values indicate the amount of data that went over his data package plan the negative values indicate the amount of data that was under the plan identify the amount of Emmanuel's use less least amount of data justify your response
Answers
Answer:
Febuary -2.25
Explanation:
We are told that the negative values indicate the amount of data that was under the plan. Therefore, the more negative the value, the less of the data Emmanuel used.
With this in mind, looking at the table we note that in the month of febuary Emmnuel used -2..25 of data (2.25 untis less than his plan ) which is the least amount of data used in any month.
connie is packing for a trip. she has 16 pairs of shoes. if she has room to pack 7 pairs, how many ways can she choose which shoes to take?
Answers
Combinatorics
If Connie had only 7 pairs of shoes and she has room for 7 pairs of shoes, then she has only one way to take her shoes.
If she had 8 pairs of shoes, then she can select any group of shoes that leaves one pair out. This makes 8 possible ways to choose.
When the number of pairs of shoes goes up, then the counting gets more complex. That is when combinatorics is a useful tool.
If we have a total of n elements to select m, where the order of selection is not important, then the total number of selections is given by:
[tex]C_{n,m}=\frac{n!}{(n-m)!\cdot m!}[/tex]Where the sign (!) is the factorial of a number.
Connie has n=16 pairs of shoes and she will take m=7 from them, thus the number of possible ways or combinations is:
[tex]\begin{gathered} C_{16,7}=\frac{16!}{(16-7)!\cdot7!} \\ C_{16,7}=\frac{16!}{(9)!\cdot7!} \end{gathered}[/tex]Expanding the factorial down to match the greatest factorial in the denominator:
[tex]C_{16,7}=\frac{16\cdot15\cdot14\cdot13\cdot12\cdot11\cdot10\cdot9!}{(9)!\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}[/tex]Simplifying and calculating:
[tex]C_{16,7}=\frac{57,657,600}{5,040}=11,440[/tex]Connie can choose in 11,440 ways the shoes to choose
Question 6 Clare volunteers at a local library during the summer. Her work includes putting labels on 750 books. How many minutes will she need to finish labeling all books if she takes no breaks and labels 15 books a minute Question 7 Suppose Clare labels the books at a constant speed of x books per minute. Write an equation that represents the relationship between her labeling speed and the number of minutes it would take her to finish labeling.
Answers
Clare is going to need 50 minutes to label 750 books
1) Gathering the data
750 books
15 books per minute
2) Let's set a proportion for that, considering the fact that her speed is constant
15 books ----------------------1 minute
750 -------------------------------x
15x = 750
x= 50 books
3) Clare is going to need 50 minutes to label 750 books.
A square box is being cut apart and has the measurements shown below. What is the area of the box ?
Answers
6 square box, each of length 3.5 inches:
Area of the original box = 6 x area of the smaller box = 6 x L x L
Area of the original box = 6 (3.5 x 3.5) = 6 (12.25) = 73.5 square inches
There are two bags containing only white and blue marbles.BagA has 11 white marbles and 9 blue marbles.Bag B has 6 white marbles and 2 blue marbles
Answers
from Least likely to Most likely
Event 3,
Event 1,
Event 4,
Event 2
STEP-BY-STEP EXPLANATION
BAG A contains:
no of white marbles = 11
no of blue marbles = 9
Total marbles in Bag A
= 11 + 9 = 20
Prob. of choosing a white marble in Bag A
= 11 / 20
Prob. of choosing a blue marble in Bag A
= 9 / 20
BAG B contains:
no of white marbles = 6
no of blue marbles = 2
Total marbles in Bag B
= 6 + 2 = 8
Prob. of choosing a white marble in Bag B
= 6 / 8
Prob. of choosing a blue marble in Bag B
= 2 / 8.
Event 1: Choosing blue marble from Bag B
= 2 / 8 = 0.25
Event 2: Choosing white marble from Bag B
= 6 / 8 = 0.75
Event 3: Choosing purple marble from Bag A
= 0 / 20 = 0
Event 4: Choosing white marble from Bag A
= 11 / 20 = 0.55.
Hence, from Least likely to Most likely we have Event 3, Event 1, Event 4, Event 2.
two hundred million thirty five
Answers
The given number is two hundred million thirty five.
As you can observe, this number has hundred millions, tens, and units. Specifically, it has 2 hundred million, 3 tens and 5 units. We write each part as a sum, as follows
[tex]200,000,000+30+5[/tex]Then we add, so the number is
[tex]200,000,035[/tex]One type of dog treat, Barker, contains 3 oz and sells for $1.95. DogYum contains 3.25 and sells for $2.10.Which is less expensive, per ounce?A. DogYum is less expensive, per ounceB. Barker is less expensive, per ounce
Answers
Solution
- We are told that dog treats, Barker and DogYum cost $1.95 for 3 oz and $2.10 for 3.25 oz respectively and we are required to find out which is more expensive per ounce.
- In order to find the more expensive dog treat, we need to find the cost per ounce for each item. The cost per ounce is given by the formula below:
[tex]\frac{\text{Cost of dog treat (In Dollars)}}{\text{Weight of dog treat (In Oz)}}[/tex]- The dog treat with the larger cost per oz is the more expensive item.
- Thus, let us calculate the cost per oz for both items.
DogYum:
[tex]\begin{gathered} \frac{\text{Cost of DogYum}}{\text{Weight of DogYum}}=\frac{2.10}{3.25}=0.646 \\ \\ \text{Thus, the cost per oz of the DogYum treat is \$0.646/oz} \end{gathered}[/tex]Barker:
[tex]\begin{gathered} \frac{\text{ Cost of Barker}}{\text{Weight of Barker}}=\frac{1.95}{3}=0.65 \\ \\ \text{Thus, the cost per oz of the Barker treat is \$0.65/oz} \end{gathered}[/tex]- We can observe that Barker is marginally more expensive than DogYum. We can confirm this by simply subtracting the cost per oz for both treats.
[tex]\text{Barker - DogYum}=0.65-0.646=0.004[/tex]Final Answer
The answer is:
DogYum is less expensive, per ounce
the answer to a multiplication problem is 3 /5 select to see if each statement is true or false
Answers
Part (A):
The value of 3/5 in the decimal form is 0.6 which is less than 1.
Thus, one of the factors of the should be less than 3/5 and other greater than 3/5.
Part (B):
For example take,
2 and 3/10.
The product of 2 and 3/10 is 3/5.
2 is greater than 3/5 and 3/10 is smaller than 3/5.
I need help with this question my answer was 2 but I for sure
Answers
Given
Distance = 30 km
Time = 1 1/2 hours = 3/2 hours
Find
rate in km per hour
Explanation
we need to find the rate in km per hour.
as we now the rate is also called a speed
so , speed = distance/time
[tex]\begin{gathered} rate=\frac{distance}{time} \\ \\ rate=\frac{30}{\frac{3}{2}} \\ \\ rate=\frac{30\times2}{3} \\ \\ rate=\frac{20km}{h} \end{gathered}[/tex]Final Answer
Therefore, the rate of ship is 20km/hr
so , the correct option is 1.
Evaluate the expression when x=5 and z=7.25z+ xSimplify your answer as much as possible.
Answers
We are given the expression
[tex]\frac{5z+x^2}{x}[/tex]and are asked to evaluate it when x=5 and z=7. For this, we simply replace the given values on the original expression, like this:
[tex]\frac{5(7)+(5)^2}{5}[/tex]And now we simplify it as much as possible:
[tex]\frac{35+25}{5}=\frac{60}{5}=12[/tex]Find the composition of transformations thatmap ABCD to EHGF.Reflect over the [? ]-axis, then translate(x+[ ], y+[ ]).Note: Enter x ory for axis.BсGH2DE124-3-2-10Enter
Answers
Let's complete the transformation from ABCD to EHGF.
Let's first determine the coordinates of the two figures.
Figure ABCD:
A : -5, 2
B : -3, 4
C : -2, 4
D : -1, 2
Figure EHGF:
E : 4, 1
H : 2, 3
G : 1, 3
F : 0, 1
Reflecting ABCD over the y - axis, we get:
P (x, y) = P' (-x, y)
For ABCD:
A : -5, 2 = A' : 5, 2
B : -3, 4 = B' : 3, 4
C: -2, 4 = C' : 2, 4
D: -1, 2 = D' : 1, 2
Let's now complete the translation,
(x + S), (y + T)
Where,
S = translation at x-axis
T = translation at y-axis
We get,
A' : 5, 2 = E : 4, 1 → S, T = 4 - 5, 1 - 2 = -1, -1
B' : 3, 4 = H : 2, 3 → S, T = 3 - 4, 3 - 4 = -1, -1
C' : 2, 4 = G : 1, 3 → S, T = 1 - 2, 3 - 4 = -1, -1
D' : 1, 2 = F: 0, 1 → S, T = 0 - 1, 1 - 2 = -1, -1
Therefore, we can say that the translation is (x + _, y + _) = (x + (-1), y + (-1)) = x - 1, y - 1
B Ε D A Ε Which two triangles are congruent? Ο ΔΑΒC ΔΕFD Ο ΔABC_ΔΕΡΕ Ο ΔΑΒC2 DEE Ο Ο ΔΑΒΟ ΔΕΡΕ
Answers
From the figure given;
Observe that, side AB is congruent to EF.
Side AC is congruenct ED
side BC is congruent to FD
From the above, we can deduce that Δ ABC ≅ Δ EFD
In AVWX, XV W X and m_W = 27°. Find mZX.
Answers
We will solve as follows:
From the information given we can deduce that the triangle is Isosceles. We also have from theorem that angles that are opposite to congruent sides are also congruent, therefore angle
Using this, and the fact that the sum o all internal angles of a triangle add to 180°, the following is true:
[tex]V+W+X=180[/tex]Now, we replace the values we know:
[tex]27+27+X=180\Rightarrow X=126[/tex]So, the value of angle
find the annual percentage yield apyA bank offers an apr of 2.5% compounded semiannually
Answers
Apr = 2.5% every 6 months
Use formula
N = N1 ( 1+ 2.5/100)^ 2
APY= (1 + r/2) ^2 - 1
. = ( 1 + 2.5/2) ^2 - 1
. = (2.25)^2 - 1
. = 4.0625
Then answer is
Annual percentage yield
APY = 4.06 %
two objects leave from point b at a right angle after ten seconds, object A has moved 12 meters and object C has moved 5 meters. what is the distance between points a and C
Answers
In order to calculate the distance between A and C, we can use the Pythagorean Theorem, since we have a right triangle:
[tex]\begin{gathered} AC^2=12^2+5^2 \\ AC^2=144+25 \\ AC^2=169 \\ AC=13 \end{gathered}[/tex]So the distance between A and C is 13 meters.
Create a real life word problem that represents a partial variation and has a constant of variation of 4
Answers
We and a problem with a constant of variation of 4.
We can think as following:
Kate wants to make a party and make candies for her guests. She intent to make at least 15 candies and an additional of 4 candies for each guest. Thetermine the total number of candies, c, that Kate must make according to the total numer of guests, g.
Answer: c = 4g + 15
what is the missing factor in the following factoring problem
Answers
notice that we have one factor of the complete expression, then, we can make the following division:
therefore, the missing factor is 7y+4
Over the past 7 days, Ms.Burge found that the temperature outside had dropped a total of 28 degrees. What is the average temperature per day?
Answers
Remember that the average or mean is the ratio between the sum of all numbers involve and the total number of items there.
In this case, the total is 28 degrees, and the total number of days is 7, we just have to divide.
[tex]\bar{x}=\frac{28}{7}=4[/tex]Therefore, the average temperature per day is 4 degrees.Stefan can pick forty bushels of apples in12 hours. One day his friend Gabriellahelped him and it only took 6.67 hours.Find how long it would take Gabriella todo it alone.
Answers
Stefan picks up 40 bushels of apples in 12 hours.
With the helo of gabriela they do so th 6.67 hours. We can represent this in the following table:
Bulshels of apples Hours
40 6.67
We need to multiply these number by 2, so that we can know how much time it would take for Gabriella to pick 40 alone.
Bulshels of apples Hours
40 6.67
80 13.34
Now, since picking 80 together takes 13.34 hours, and we know that tefan picks up 40 in 12 hours, the other 40 where picked up by Gabriella in:
[tex]13.34-12=1.34[/tex]Answer: 1.34 Hours
The function y =30+5x represents the cost y (in dollars) of having your dog groomed and buying x for extra services . can u please just do C (graph the function using six values of it's domain
Answers
Problem
The function y =30+5x represents the cost y (in dollars) of having your dog groomed and buying x for extra services . can u please just do C (graph the function using six values of it's domain
Solution
Part a
For this case a linear function is given by: y = mx+b
If we compare the function y =30 +5x we can see that we have the sam efunctional form and we can conclude that yes is a linear function with m = 5 and b =30
Part b
For this part since we have a linear function the domain is given by:
D= [0,1,2,3,4,5}
Then the domain is discrete
Part c
For this case we have the following solution:
the recursive rule is a1=__and a n-1 for n> __.the explicit rule is a n= ___(__)n-1
Answers
We are required to find both the recursive rule and the Explicit rule for the values given in the table.
Recursive rule:
To get the recursive rule, we need to find the common ratio.
The common ratio (r) is found by dividing the 2nd term by the 1st term, or the 3rd term by the 2nd term, or the 4th term by the 3rd term, and so on.
We can generalize and say the common ratio is also the division of the nth term by the (n-1)th term.
Let us put these facts into mathematical expressions:
[tex]\begin{gathered} Let,_{}_{} \\ a_1=\text{first term} \\ a_2=\text{Second term} \\ a_3=\text{third term} \\ \text{...} \\ a_n=\text{nth term} \\ \\ \text{common ratio (r)=}\frac{a_1}{a_2}=\frac{a_3}{a_2}=\frac{a_4}{a_2}\ldots=\frac{a_n}{a_{n-1}} \\ \\ \text{if a}_1=1,a_2=3_{} \\ \therefore r=\frac{a_2}{a_1}=\frac{3}{1}=3 \end{gathered}[/tex]Thus, we can now write the recursive formula as:
[tex]\begin{gathered} \frac{a_n}{a_{n-1}}=\text{common ratio (r)} \\ \\ \therefore\frac{a_n}{a_{n-1}}=3 \\ \\ \therefore a_1=1,a_n=3a_{n-1}\text{ for n}\ge1 \end{gathered}[/tex]Explicit rule:
To get the rule, we need to recognize the pattern:
[tex]\begin{gathered} \text{when n = 1} \\ a_1=1=3^0 \\ \\ \text{When n = 2} \\ a_2=3=3^1 \\ \\ \text{When n= 3} \\ a_3=9=3^2 \\ \\ \text{When n= 4} \\ a_4=27=3^3 \\ \\ \text{When n = 5} \\ a_5=81=3^4 \end{gathered}[/tex]We can see a pattern developing and from that, we can generalize:
[tex]\begin{gathered} a_1=3^0 \\ a_2=3^1 \\ a_3=3^2 \\ a_4=3^3 \\ a_5=3^4 \\ \ldots \\ a_n=3^{n-1} \end{gathered}[/tex]Thus, the explicit rule is:
[tex]\begin{gathered} a_1=1 \\ a_n=1\times3^{n-1} \end{gathered}[/tex]find all of the zeros of p (x) = x^3-x^2+2, given that 1-i is a zero. (if there is more than one zero, separate them with commas.)
Answers
We have the next polynomial function
[tex]P(x)=x^3-x^2+2[/tex]In order to find the zero we punt the function equal to zero
[tex]x^3-x^2+2=0[/tex]We have a given zero that is 1-i
x=1-i
than means that we need the conjugate of the zero given is also a zero
x=1+i
and then we need a third zero because the polynomial has third-degree it can be calculated if we factorize the polynomial given
[tex]\mleft(x-1-i\mright)\mleft(x-1+i\mright)\mleft(x+1\mright)=0[/tex]Therefore the zeros are
x=-1
x=1-i
x=1+i
The vertex of this parabola is at (-2,5). Which of the following could be its equation? (-2,5) -5 O A. y = 3(x-2)2-5 O B. y = 3(x-2)2 + 5 O C. y= 3(x+ 2)2-5 BE E EIE O D. y=3(x + 2)2 + 5 anter
Answers
The base equation of a parabola is F(X) = X^2 + C
if we make F(X+2) it means the chart will move 2 units to the left
on the other hand, if the make F(X) + 5, it means the chart will displace 5 units upwards
As a result, from the given chart, our equation would be: y = 3(x+2)^2 + 5
What is the equivalent decimal of 16/52? Enter your answer, rounded to the nearest thousandth of a degree, in the box.
Answers
This is a graph of the relationship between millimeters of rainfall and umbrella sales. In your own words, explain what happens to umbrella sales as the amount of rainfall changes.
Answers
On the graph, the x axis represents rainfall while the y axis represents number of umbrella sold. We can see that as the amount of rainfall is increasing, the number of umbrellas sold is increasing. This is shown by the line moving upwards in the positive direction. the relationship between the amount of rainfall and the number of a umbrellas sold is a positive association.
Question 1 (Essay Worth 10 points)(03.03 MC)A biologist is studying the growth of a particular species of algae. She writes the following equation to show the radius of the algae, f(d), in mm, after d days:f(d) = 11(1.01)dPart A: When the biologist concluded her study, the radius of the algae was approximately 11.79 mm. What is a reasonable domain to plot the growth function? (4 points)Part B: What does the y-intercept of the graph of the function f(d) represent? (2 points)Part C: What is the average rate of change of the function f(d) from d = 2 to d = 7, and what does it represent? (4 points) Source StylesFormat ◢ Question 2 (Essay Worth 10 points)(03.02, 03.03, 03.04 MC)The amount of money in an account may increase due to rising stock prices and decrease due to falling stock prices. Marco is studying the change in the amount of money in two accounts, A and B, over time.The amount f(x), in dollars, in account A after x years is represented by the function below:f(x) = 1,264(1.09)xPart A: Is the amount of money in account A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)Part B: The table below shows the amount g(r), in dollars, of money in account B after r years:r (number of years)1234g(r) (amount in dollars)1,3751,512.501,663.751,830.13Which account recorded a greater percentage change in amount of money over the previous year? Justify your answer. (5 points)
Answers
A biologist studying the growth of a particular species of algae and later designed a model equation for the growth
F(d) = 11(1.01)d
Where, f(d) is in mm and d is days
QUESTION A
F(d) = 11(1.01)d
f(d) = 11.79 mm
Substitute f(d) = 11.79 mm into the model equation
11.79 = 11(1.01)d
11.79 = 11 * 1.01 * d
11 .79 = 11.11 x d
11.79 = 11.11d
Divide both sides by 11.11
11.79/11.11 = 11.11d/11.11
d = 11.79/11.11
d = 1.061
Thde reasonable domain to plot the growth function is 1.061 < d > 1.061
Situation:Holly wants to save money for anemergency. Holly invests $1,000 in anaccount that pays an interest rate of6.25%
Answers
Given data:
Amount of money invested, P = $1000
Interest rate, r = 0.0625
Total money in the account, A = 5500
Now, to find the years use simple interest rate formula that is
[tex]A=P(1+rt)[/tex]Therefore, t will become
[tex]t\text{ = (A/P}-1\text{)/r}[/tex]Putting the values we get,
[tex]t=(\frac{5500}{1000}-1)\text{ / 0.0625}[/tex][tex]\begin{gathered} t=(5.5-1)\text{ / 0.0625} \\ t=\frac{4.5}{0.0625} \\ t=72 \end{gathered}[/tex]Thus, it will take 72 years for the account to reach 5500.