A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into three identical smaller rectangular plot by placing to fence is parallel to the field shorter side. Find the dimensions that maximize the enclosed area. Write your answer as a fraction reduced to lowest term
Answers
The diagram of the problem is:
S is the length of the shorter side of the fence. L is the length of the longest side of the field.
We know that the perimeter of the rectangle is 800ft. This means:
[tex]2S+2L=800[/tex]And the area:
[tex]A=SL[/tex]The smaller rectangles will have dimensions:
The area is:
[tex]a=\frac{SL}{3}[/tex]As we can see, if we maximize the area of the bigger rectangle "A", we are also maximizing the area of the smaller rectangles "a".
Then, we have two equations:
[tex]\begin{gathered} 2S+2L=800 \\ A=SL \end{gathered}[/tex]We can solve for L in the first equation:
[tex]\begin{gathered} 2S+2L=800 \\ 2L=800-2S \\ L=400-S \end{gathered}[/tex]Then substitute in the second:
[tex]A=S(400-S)[/tex]Simplify:
[tex]A=400S-S^2[/tex]This is a function of the area depending on the length of the shorter side of the rectangle:
[tex]A(S)=400S-S^2[/tex]We can find the maximum of this function if we find the value where the derivative of this function is 0.
Let's differentiate:
[tex]A^{\prime}(S)=400-2S[/tex]And now we find where A'(S) = 0:
[tex]\begin{gathered} 0=400-2S \\ 2S=400 \\ S=200 \end{gathered}[/tex]We have found that the shorter side must have a length of 200ft to maximize the area. Let's find the length of the larger side:
[tex]L=400-200=200[/tex]As expected, the quadrilateral which maximizes the area is the square. Thus, the dimensions of the field are 200ft x 200ft
Related Questions
Find WZ if WX = 4a, XY = 28, YZ = 6a - 4 and XZ = 10a + 12
Answers
1) In this line segment, we can apply the the addition postulate to solve it and write:
WZ = WX +XY +YZ
10a +12 = (4a +28) +(6a - 4)
10a +12= 10a +24
0a = 12 No solution
2) Since this is a non-solution equation, then we can't solve for a and find the length of WZ
A pair of linear equations is shown:
y = −3x + 5
y = x + 2
Which of the following statements best explains the steps to solve the pair of equations graphically?
On a graph, find the point of intersection of two lines; the first line has y-intercept = 5 and slope = −3, and the second line has y-intercept = 2 and slope = 1.
On a graph, find the point of intersection of two lines; the first line has y-intercept = −3 and slope = 5, and the second line has y-intercept = 1 and slope = 2.
On a graph, find the point of intersection of two lines; the first line has y-intercept = −5 and slope = 3, and the second line has y-intercept = −2 and slope = −1.
On a graph, find the point of intersection of two lines; the first line has y-intercept = 3 and slope = −5, and the second line has y-intercept = −1 and slope = −2.
Answers
The graph that shows the solution to the pair of linear equations is shown below. The best explanation for the step to take is: A. the first line is the red line while the second line is the blue line.
How to Solve a Pair of Linear Equations Graphically?To find the solution to a pair of linear equations using a graph, plot the given equations on a graph using their slopes and their y-intercepts.
The point where the two lines intersect is the solution to the pair of linear equations.
Given the linear equations:
y = −3x + 5
y = x + 2
The graph of y = -3x + 5 has a slope of -3 and a y-intercept of 5, while the graph of y = x + 2 has a slope of 1 and a y-intercept of 2.
The diagram that shows the graph where the two lines intersect at point (0.75, 2.75) is shown below.
The solution is: (0.75, 2.75)
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Solve: x−6−−−−−√=x−6 . Enter the exact answers. The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2;4;6 or x+1;x−1 ). The order of the list does not matter.
Answers
ANSWER:
x = 6; 7
STEP-BY-STEP EXPLANATION:
We have the following equation:
[tex]\sqrt{x-6}=x-6[/tex]We solve for x:
[tex]\begin{gathered} \left(\sqrt{x-6}\right)^2=\left(x-6\right)^2 \\ \\ x-6=x^2-12x+36 \\ \\ x^2-12x-x+36+6=0 \\ \\ x^2-13x+42=0 \\ \\ -13x=-6x-7x \\ \\ x^2-6x-7x+42=0 \\ \\ \left(x^2-6x\right)+\left(-7x+42\right)=0 \\ \\ x\left(x-6\right)-7\left(x-6\right) \\ \\ \left(x-6\right)\left(x-7\right)=0 \\ \\ x-6=0\rightarrow x=6 \\ \\ x-7=0\operatorname{\rightarrow}x=7 \\ \\ \text{ We check each solution, as follows:} \\ \\ \sqrt{6-6}=6-6\rightarrow0=0\rightarrow\text{ True} \\ \\ \sqrt{7-6}=7-6\rightarrow1=1\rightarrow\text{ True} \\ \\ \text{ Both solutions are correct therefore:} \\ \\ x=6;7 \end{gathered}[/tex]One bag of rice that weighs 3.5 pounds will have enough rice if a recipe calls for 54 ounces of rice. (1 pound = 16 ounces) (3 points)
Answers
One bag of rice that weighs 3.5 pounds will have enough rice for the recipe if a recipe calls for 54 ounces of rice . the given statement is True .
In the question ,
it us given that
weight of one bag of rice = 3.5 pounds
given that ,
1 pound [tex]=[/tex] 16 ounces
So, we need to convert 3.5 pounds to ounces first .
we have 3.5 pounds = 16×3.5 ounces
= 56 ounces .
So , yes there will be enough rice for the recipe as only 54 ounces are needed and we have 56 ounces , that is 2 extra ounces of rice .
Therefore , "One bag of rice having weight 3.5 pounds ,will have enough rice for the recipe if a recipe calls for 54 ounces of rice" . the given statement is True .
The given question is incomplete , the complete question is
One bag of rice that weighs 3.5 pounds will have enough rice if a recipe calls for 54 ounces of rice. (1 pound = 16 ounces) . The given statement is TRUE or FALSE ?
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help me with the last question pleaseeeeeeeeeeeeeee
Answers
Answer:
Answer:
d
Step-by-step explanation:
The number of years must be non-negative.
This eliminates all of the options except for d.
Simplify using exponent B-³
Answers
Answer:
B^-3
1/B³
Step-by-step explanation:
please rate as brainliest
Dillon went out for dinner with four friends. All had the same meal and dessert, except for Dillon who did not order the $5 dessert. The total bill came to $45. Write an equation to determine how much each of his friends paid for their meal.
Answers
The equation to determine each of his friends paid for their meal would be; 4x + x - 5 = 45.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given that the Dillon and 4 friends went for dinner.
They all had the same meal except Dillion who did not order the 5 dollars dessert.
The Total bill paid = $45
Let the amount paid by each friend = x
Then Amount paid by dillion = x - 5
4x + x - 5 = 45
5x - 5 = 45
5x = 45 + 5
5x = 50
x = 10
Thus, the Amount paid by each of the 4 friends = $10
Hence, Amoubt paid by Dillion is $10 - $5 = $5
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X-2y=5 and 2x+3y=10
Answers
Hello!
Let's set up some things:
[tex]x-2y=5[/tex] <-- equation 1
[tex]2x+3y = 10[/tex] <-- equation 2
________________
(equation 1) multiplied by 2
[tex]2x - 4y=10[/tex] <-- equation 3
________________
(equation 2) - (equation 3)
7y = 0
y = 0 <-- equation 4
________________
(equation 4)'s value of y into (equation 1)
[tex]x-2(0) = 5\\x = 5[/tex]
Thus x = 5 and y = 0
Hope that helps!
A standard sheet of paper in Europe is `29.7` centimeters long and `21` centimeters wide.
Which standard sheet of paper has a greater area?
Answers
Standard sheet of paper in Europe has a greater area i.e. 29.7 cm. long and 21 cm wide.
1 in = 2.54 cm (in Europe)
In SI units standard-sized paper is 21.0 cm by 29.7 cm, which is equivalent to 8.268 in by 11.693 in.
In U.S.
U.S. sized standard paper is 8.0 in by 11.0 in therefore the SI (internationally) sized paper is longer.
Standard sheet of paper in Europe has a greater area than standard sheet of paper in U.S.
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Carina has a piece of wire that is 301cm long . She bends the wire to make 6 shapes of the same perimeter . She has 97cm of wire left . What is the perimeter of each shape ?
Answers
Answer:
34
Step-by-step explanation:
301 - 97 = 204
204 / 6 = 34
Angle 1,7,3 and 5 all those angles has the same equivalent to the angle 5 angle 2 is supplementary to angle 1 so angle 2 is 30 degrees because 150+ 30 is 180 degrees and we use the same thing that I did in the first step that if angle number to is 30 angle 8,4 and 6 is going to be 30
Answers
Given:-
To find all angle values.
So from the given image. we get,
[tex]\begin{gathered} \angle5+\angle4=180 \\ \angle7+\angle4=180 \\ \angle7+\angle6=180 \\ \angle6+\angle5=180 \end{gathered}[/tex]Also,
[tex]\begin{gathered} \angle1=\angle7 \\ \angle2=\angle4 \\ \angle8=\angle6 \\ \angle3=\angle5 \end{gathered}[/tex]So now we know angle 5 is 150 degrees. so we get,
[tex]\begin{gathered} \angle5+\angle4=180 \\ 150+\angle4=180 \\ \angle4=180-150 \\ \angle4=30 \end{gathered}[/tex]So angle 4 is 30 degree.
Also,
[tex]\begin{gathered} \angle7+\angle4=180 \\ \angle7+30=180 \\ \angle7=150 \end{gathered}[/tex]So angle 7 is 150 degree.
Also,
[tex]\begin{gathered} \angle6+\angle5=180 \\ \angle6+150=180 \\ \angle6=30 \end{gathered}[/tex]So angle 6 is 30 degree.
So now we equate the values. so we get,
[tex]undefined[/tex]Please help with this word problem,Laurie, Moesha, and Carrie went to the mall. Laurie spent $20, Moesha spent $25, and Carrie spent $30. How many did they spend in all?
Answers
To determine how much they all spent we need to add the expenses of each person: Laurie, Moesha, and Carie. This is shown below:
[tex]\begin{gathered} \text{ total=Laurie+Moesha+Carrie} \\ \text{total}=20+25+30=75 \end{gathered}[/tex]They spent a total of $75.
At many bakeries, there are deals that if you buy twelve donuts, you get a thirteenth one at a discount. Sometimes "13" items is called a baker's dozen. I One store is advertising: Donuts! $1.05 individually, or $13 per baker's dozen. They want to program their cash register to be able to show the exact price for any number of donuts. You will help them by producing a data table, a graph and an algorithm. You will: 1. Fill in and expand the data table 2. Make a scatter plot to show the pattern visually. This means to plot an (x,y) point for every row in the data table. For example the first point can be (1, 1.05) 3. Develop an algorithm to find the cost for any number of donuts. Note: An algorithm is like a set of instructions to solve a problem. They are useful for telling computers or people exactly how to do something. For example, here is an algorithm that finds 15% of a number: Step 1: take any number, callit N.
Answers
Table
Number of donuts Price
1 $1.05
2 $ 2.1
3 $3.15
4 $4.2
5 $5.25
6 $6.3
10 $10.5
slope = (10.5 - 1.05) / 10 - 1
= 9.45 / 9
= 1.05
Equation
y - 10.5 = 1.05 (x - 10)
y = 1.05x - 10.5 + 10.5
y = 1.05x
For 40 donuts
y = 1.05(40)
y = $42
Write an equation in standard form for the line that passes through the given points.
(1, 2) and (5, 7)
Answers
The equation of the line in standard form is represented by - (5 / 3) · x + (4 / 3) · y = 1.
What is the equation of the line that passes through two points?
In this problem we must derive the equation of the line in standard form that passes through the two points. The standard form is described below:
a · x + b · y = 1
Then, we must solve for a and b to determine the coefficients of the line equation:
a + 2 · b = 1
5 · a + 7 · b = 1
By numerical methods, the solution to the system is:
a = - 5 / 3, b = 4 / 3
The equation in standard form - (5 / 3) · x + (4 / 3) · y = 1.
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HELP
QUICKOU#FR{PIEWuf'oieud'f'doifu
Answers
Apply the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, a^2 + b^2 = c^2 where c is the hypotenuse.
a^2 + b^2 = c^2
8^2 + 5^2 = c^2
64 + 25 = c^2
89 = c^2
√89 = √c^2
c = √89
A police car traveling south toward Sioux Falls, Iowa, at 160 km/h pursues a truck traveling east away from Sioux Falls at
140 km/h.
At time t = 0, the police car is 60 km north and the truck is 50 km east of Sioux Falls.
Calculate the rate at which the distance between the vehicles is changing at t = 10 minutes.
(Use decimal notation. Give your answer to three decimal places.)
Answers
Using the Pythagorean Theorem, the rate at which the distance between the vehicles is changing at t = 10 minutes is of -20 km/hour.
What is the Pythagorean Theorem?The Pythagorean Theorem relates the length of the legs [tex]l_1[/tex] and [tex]l_2[/tex] of a right triangle with the length of the hypotenuse h, stating that the hypotenuse squared is the sum of the legs squared, according to the equation given below:
[tex]h^2 = l_1^2 + l_2^2[/tex]
In the context of this problem, the distances of the cars from Sioux Falls, considering the initial distances and the velocities, are given as follows:
Police: P(t) = 60 - 160t.Truck: T(t) = 50 + 140t.The distance after t hours is the hypotenuse of a right triangle in which the legs are the functions, hence:
D²(t) = P²(t) + T²(t).
Hence the rate of change of the distance after t hours is:
2D'(t) = 2P'(t) + 2T'(t)
Simplifying by 2:
D'(t) = P'(t) + T'(t).
Applying the exponent rules, the derivatives are given as follows:
P'(t) = -160.T'(t) = 140.Hence the constant rate is:
D'(t) = -160 + 140 = -20 km/hour.
Meaning that the police car is getting 20 km closer to the truck each hour.
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Please find it if needed.-The Area of a Rectangle-The Area of a Triangle-The Area of a Square-Or The Area of a Circleand find the area of the composite figure (andplease put the areas separate)FIND THE AREA OF THE SHADED REGION
Answers
First find the area of the rectangle
A = l*w = 9*16 = 144
The unshaded area is a trapezoid
A = 1/2 ( b1+b2) *h = 1/2 ( 5+11) * 4 = 1/2 (16) * 4 =32
The area of the shaded region is
rectangle - trapezoid
144-32
112cm^2
Answer:
112cm²
Step-by-step explanation:
First, let us find the area of the rectangle.
Area = length × width
Area = 16 cm × 9 cm
Area = 144 cm²
Now, let us find the area of the trapezium.
Area = 1/2 ( Sum of the parallel sides ) × height
Area = 1/2 × ( 11 + 5 ) × 4
Area = 1/2 × 16 × 4
Area = 32 cm²
Area of the shaded region.
To find the area of the shaded region, subtract the area of the rectangle from the area of the trapezium.
Area = 144cm² - 32cm²
Area = 112cm²
Part 1: Factorial!2. Explain the steps that you took to calculate it on your calculator.
Answers
Given:
a factorial 14! is given
Find:
we have to evaluate the given factorial
Explanation:
we know, the factorial 14! is calculated as following
14! = 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
14! = 87178291200
or
14! = 8.71783E10
when trying to find the probability of rolling a number four on a number cube with the number 1 through 6 how many desired outcomes are there
Answers
ANSWER
One desired outcome
EXPLANATION
We are trying to roll the number 4 on a number cube with numbers 1 through 6.
Since we are only trying to get one outcome/result from the experiment (4), there is only one desired outcome.
The answer is 1.
Find the area of quadrilateral ABCD. [Hint: the diagonal divides the quadrilateral into two triangles.]A. 28.53 units²B. 26.47 units²C. 27.28 units²D. 33.08 units²
Answers
Answer
A. 28.53 units²
Explanation
Finding the area of irregular quadrilateral ABCD, we divide the given figure into shapes (two triangles) as shown below:
Then, we find the area of the two triangles.
Triangle ABD:
[tex]\begin{gathered} Area=\sqrt{s(s-a)(s-b)(s-c)} \\ \\ s=\frac{a+b+c}{2}=\frac{2.89+8.59+8.6}{2}=\frac{20.08}{2}=10.04 \\ \\ Area\text{ }of\text{ }triangle\text{ }ABD=\sqrt{10.04(10.04-2.89)(10.04-8.59)(10.04-8.6)} \\ \\ Area\text{ }of\text{ }triangle\text{ }ABD=\sqrt{10.04(7.15)(1.45)(1.44)} \\ \\ Area\text{ }of\text{ }triangle\text{ }ABD=\sqrt{149.889} \\ \\ Area\text{ }of\text{ }triangle\text{ }ABD=12.24\text{ }unit^2 \end{gathered}[/tex]Triangle ADC:
[tex]\begin{gathered} Area=\sqrt{s(s-a)(s-b)(s-c)} \\ \\ s=\frac{a+b+c}{2}=\frac{4.3+7.58+8.6}{2}=\frac{20.48}{2}=10.24 \\ \\ Area\text{ }of\text{ }triangle\text{ }ADC=\sqrt{10.24(10.24-4.3)(10.24-7.58)(10.24-8.6)} \\ \\ Area\text{ }of\text{ }triangle\text{ }ADC=\sqrt{10.24(5.94)(2.66)(1.64)} \\ \\ Area\text{ }of\text{ }triangle\text{ }ADC=\sqrt{265.346} \\ \\ Area\text{ }of\text{ }triangle\text{ }ADC=16.29\text{ }unit^2 \end{gathered}[/tex]Therefore, the area of the quadrilateral ABCD = the Sum of the two triangles
The area of the quadrilateral ABCD = 12.24 units² + 16.29 units² = 28.53 units²
Please help with this question pl
Answers
The missing reason 6 of the two column proof that m∠Z = 65° is; Substitution
How to carry out two column proof?The two column proof is as follows;
Statement 1: XYZ is a triangle
Reason 1; Given
Statement 2: m∠x + m∠y + m∠z = 180°
Reason 2: Postulate
Statement 3: m∠y = 50°
Reason 3: Given
Statement 4: ∠X ≅ ∠Z
Reason 4: Given
Statement 5: m∠X = m∠Z
Reason 5: Congruency
Statement 6: m∠Z + m∠Z = 180°
Reason6: Substitution
Statement 7: m∠Z + m∠Z = 130°
Reason 7: ?
Statement 8: 2(m∠Z) = 130°
Reason 8: Algebra
Statement 9: m∠Z = 65°
Reason 9: Division property of equality
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Х m . In the segment shown point M is the mi M the midpoint of AB. Given AM = 3x+ 12 and MB = MB = 5x -4, find the length of AM. DP A M B M
Answers
From the information provided, M is the midpoint of line segment AB. This implies that the segments AM and MB are two equal halves of the entire length,
Therefore, we would have the following;
[tex]\begin{gathered} AM+MB=AB \\ \text{Also,} \\ AM=MB \\ \text{Where;} \\ AM=3x+12,MB=5x-4 \\ We\text{ now have;} \\ 3x+12=5x-4 \\ \text{Collect all like terms;} \\ 12+4=5x-3x \\ 16=2x \\ \text{Divide both sides by 2;} \\ \frac{16}{2}=\frac{2x}{2} \\ 8=x \end{gathered}[/tex]Where AM = 3x+12, we now have;
[tex]\begin{gathered} AM=3x+12 \\ AM=3(8)+12 \\ AM=24+12 \\ AM=36 \end{gathered}[/tex]ANSWER:
Segment AM = 36
If one mini Hershey bar weighs 0.3 ounces how many mini candy bars would you have to eat to eat the same amount as a regular size Hershey bar that weighs 1.55 ounces?
Answers
0.3x=1.55 from here you solve for x. Do this by diving 1.55 by 0.3 to get c alone
x=5.16666 repeating
Find the area of the surface of a sphere of radius r, which is generated when rotating a semicircle centered at the origin, is rotated about its diameter.
Answers
The surface area of a sphere of radius r is
PEOPLE WHO ARE GOOD AT MATH CAN YOU PLEASE HELP ME!!!
Answers
The cost of using 19 HCF of water is $32.49
Given in the question:
The monthly cost (in dollars) of the water use (in dollars) is a linear function of the amount of water used (in hundreds of cubic feet, HCF)
The cost for using 17 HCF of water is using $32.13
and, the cost of using 35 HCF is $61.83.
To find the cost of using 19 HCF of water.
Now, According to the question:
The cost for using 17 HCF of water is $32.13
and, the cost of using 35 HCF is $61.83.
To find the slope:
(17, 32.13) and (35, 61.83)
Slope = (61.83 - 32.13)/ (35 - 17) = 1.65
We know that:
Formula of slope :
y = mx + b
32.13 = 1.65 x 17 + b
b = 1.14
The equation will be :
C(x) = 1.65x + 1.14
Now, To find the cost of using 19 HCF of water.
C(19) = 1.65 × 19 + 1.14
C(19) = $32.49
Hence, the cost of using 19 HCF of water is $32.49.
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What is the value of (-3)-4?
Answers
Answer:
-7
Step-by-step explanation:
-3-4=-7
:]
Answer:
-7
Step-by-step explanation:
Wich is an example of multiplicative identityA. a×1=aB. a+0=aC. a+(-a)=0orD. a×1/a=1
Answers
Multiplicative Identity:
The multiplicative identity property states that if a number is multiplied by 1, the result is that number.
Examples:
[tex]\begin{gathered} a\times1=a \\ 5\times1=5 \\ 20\times1=20 \\ y\times1=y \end{gathered}[/tex]Therefore, the example of multiplicative identity is option A
[tex]a\times1=a[/tex]Evaluate (2 +40) : 2 when z = 8 answer A 8 B 5C 6D 48
Answers
1) Let's solve the equation given the condition
[tex]\frac{\mleft(x+40\mright)}{2^3}\Rightarrow\frac{(8+40)}{8}\Rightarrow\frac{48}{8}=6[/tex]After plugging into the equation, all that's left is to solve it with the PEMDA order of operation.
2) So the answer is C
1233558844+12254788-1255+77777
Answers
1,233,558,844 + 12,254,788 − 1,255 + 77,777
= 1,245,890,154
Answer:
1245890154 easy, lol
I know this wasn't a real question but if it was good luck, lol
Suppose that the relation T is defined as follows.T = {(0,6), (2, -8), (-8, 2), (-3, -3)}What is the domain and range?
Answers
Given the follwing relation T:
[tex]T=\mleft\lbrace(0,6\mright),(2,-8),(-8,2),(-3,-3)\}[/tex]to write the domain, we have to gather all the first components of each ordered pair. In this case, the domain is:
[tex]DomT=\mleft\lbrace0,2,-8,-3\mright\rbrace[/tex]the range will be the second component of each ordered pair:
[tex]Range(T)=\mleft\lbrace6,-8,2,-3\mright\rbrace[/tex]