A rectangular field is to be bounded by a fence on 3 sides and a straight stream on the 4th side. Find the dimensions of the field with maximum area that can be enclosed using 1000 ft of rope. This is in the applied maximum and minimum chapter btw.
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[tex]\bf \stackrel{area}{A}=length\cdot width\implies A(w)=(1000-2w)w \\\\\\ \boxed{A(w)=1000w-2w^2} \\\\\\ \cfrac{dA}{dw}=1000-4w\implies 0=1000-4w\implies 4w=1000 \\\\\\ w=\cfrac{1000}{4}\implies \boxed{w=\stackrel{width}{250}} \\\\\\\ [1000-2(250)]\implies 1000-500\implies \boxed{l=\stackrel{length}{500}}[/tex]
and if you run a first-derivative test on say x = 249 and x = 251, you'll notice the slope is positive and negative respectively, meaning, is a maximum.
To maximize the area of the rectangular field, express the area in terms of one variable, derive the area function, and set it to zero for solving. The sides of the field yielding the maximum area are obtained using these calculations.
Explanation:This question involves optimization in calculus, specifically related to finding the maximum area of a rectangle. In such a problem, the first step is to express the area (A) of the rectangle in terms of one variable. Given a fence that is 1000 ft in length, if we name the side perpendicular to the stream as 'x', and the side parallel to it (bounded by the fence) as 'y', the perimeter of the field is 2x + y = 1000.
We can express y in terms of x: y = 1000 - 2x. The area of the rectangle, A = x*y = x*(1000 - 2x).
To find the maximum area, we find the first derivative of A with respect to x, set it to zero and solve for x. The second derivative test will confirm if it's a maximum. Based on the solutions, the maximum area is achieved with the dimensions of the rectangular field being specific values of x and y that you compute.
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Related Questions
Ollie used 276 bricks to make 6 fence columns. He used an equal number of bricks to make each column. How many bricks did he use to make each column?
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Answer: The answer is 46!
Step-by-step explanation:
Kelly drives 448 miles on the highway to visit her xousin. She uses cruise control to drive at a constant speed . Kelly travels 192 miles in the first 3 hours. At that rate, how long will the trip take?
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192 miles divided by 3 hours = 64 miles per hour
448 miles / 64 miles per hour = 7 hours total for the trip
Answer:
Trip will take 7 hours to complete.
Step-by-step explanation:
Kelly drives 192 miles in the first 3 hours.
Speed of Kelly to cover 192 miles = [tex]\frac{\text{Distance}}{\text{Time}}[/tex]
= [tex]\frac{192}{3}[/tex]
= 64 miles per hour
Kelly travels with the same speed to cover the whole distance.
Therefore, time taken to cover 448 miles = [tex]\frac{\text{Distance}}{\text{Speed}}[/tex]
= [tex]\frac{448}{64}[/tex]
= 7 hours
Kelly will meet his cousin after 7 hours.
Use implicit differentiation to find the points where the circle defined by x^2+y^2−6x−4y=-4 has horizontal and vertical tangent lines. List your answers as points in the form (a,b).
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So 2x-6=0 and x=3. To find y we solve 9+y²-18-4y=-4, y²-4y-5=0=(y-5)(y+1), so the points are (3,5) and (3,-1).
2xdx/dy+2y-6dx/dy-4=0. When dx/dy=0 we have a vertical tangent, so 2y=4, y=2, and x²+4-6x-8=-4, x²-6x=0, x=0 and 6.
The points are (0,2) and (6,2).
Which of the following slopes show that the set of points C(1, 1), D(3, -4), E(5, 8) are not collinear?
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Answer:
C (-1, 1/2, -3/5) is INCORRECT and I know it isn’t (1/3, -5/3, -2/5)...... but I’m pretty sure the answer is 1/2, 3, 5
Step-by-step explanation:
Two parallel lines are crossed by a transversal. What is the value of g? g = 75 g = 80 g = 100 g = 105
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we know that
If the lines u and w are parallel
then
∠g=[tex]105\°[/tex] ------> by alternate exterior angle
therefore
the answer is
∠g=[tex]105\°[/tex]
What does the equation x = 6 represent in the set of real numbers2?
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for a y-intercept at -3, the line crosses at (0,-3).
.
Slope, can be determined from two points:(2,0) and (0,-3)
slope (m) = (0-(-3))/(2-0) = 3/2
y-intercept at -3
slope is 3/2
.
plugging this into the "slope-intercept" form:
y = mx + b
we have:
y = (3/2)x - 3
63 is 75% of what number?
Enter your answer in the box.
Answers
(63/75 * 100)
Hope this helps!
Read the word problem below. A parking garage charges $3 for each of the first three hours. After three hours, the charge is $7 an hour. What is the total cost for five hours of parking? Which strategy that would help solve this word problem?
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$7 x 2hrs = $14
$9 + $14 = $23
The total cost would be $23
The best strategy would be to multiply the money and hours and add the totals.
Hope this helps! Let me know if you need any further assistance!
Then there would be 2 hours left so do 7*2 = 14
So this would be how it looks with no words
3*3 + 7*2 =
Then you do 14 + 9 = 23
So your answer is $23
Hope I helped :)
A line passes through the points (2, –2) and (–6, 2). The point (a, –4) is also on the line. What is the value of a? mc023-1.jpg
Answers
Answer:
[tex]a=6[/tex]
Step-by-step explanation:
We have been given coordinates of three points on the same line and we are asked to find the value of 'a'.
First of all, we will find the equation of the line in slope-intercept form of equation [tex]y=mx+b[/tex].
Let us find slope of our given line using coordinates of point [tex](2,-2)[/tex] and [tex](-6,2)[/tex].
Upon substituting the coordinates of the given points in slope formula we will get,
[tex]m=\frac{2--2}{-6-2}[/tex]
[tex]m=\frac{2+2}{-8}[/tex]
[tex]m=\frac{4}{-8}[/tex]
[tex]m=-\frac{1}{2}[/tex]
Now, we will substitute [tex]m=-\frac{1}{2}[/tex] and coordinates of point [tex](2,-2)[/tex] in slope-intercept form of equation to find the y-intercept.
[tex]-2=-\frac{1}{2}*2+b[/tex]
[tex]-2=-1+b[/tex]
[tex]-2+1=-1+1+b[/tex]
[tex]-1=b[/tex]
Therefore, the equation of line passing through the given points is [tex]y=-\frac{1}{2}x-1[/tex].
To find the value of 'a' we will substitute coordinates of point [tex](a,-4)[/tex] in our equation as:
[tex]-4=-\frac{1}{2}*a-1[/tex]
[tex]-4+1=-\frac{1}{2}*a-1+1[/tex]
[tex]-3=-\frac{1}{2}*a[/tex]
Upon multiplying both sides of our equation by -2, we will get
[tex]-3*-2=-\frac{1}{2}*a*-2[/tex]
[tex]6=a[/tex]
Therefore, the value of a is 6.
Answer:
A = 6
Step-by-step explanation:
I think this is late but i got this right hope this helps :D
If m and n are integers is 6m^2+34n-18 an even integer?
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For any integers m and n, [tex]\(6m^2 + 34n - 18\)[/tex] is always even Integer . One solution is [tex]\(m = 1, n = 2\)[/tex].
To determine if the expression [tex]\(6m^2 + 34n - 18\)[/tex] is an even integer for any integers m and n, let's analyze the terms.
The expression can be factored as [tex]\(2(3m^2 + 17n - 9)\).[/tex] Since 2 is a factor of the expression, it implies that the entire expression is even. Therefore, for any integers m and n, [tex]\(6m^2 + 34n - 18\)[/tex] is an even integer.
Now, to find possible values for m and n that satisfy the condition, consider that [tex]\(3m^2 + 17n - 9\)[/tex] must be an integer. We can choose arbitrary values for m and n and verify that [tex]\(3m^2 + 17n - 9\)[/tex]is an integer. For example, let m = 1 and n = 2:
[tex]\[3(1)^2 + 17(2) - 9 = 3 + 34 - 9 = 28\][/tex]
Since 28 is an integer, m = 1 and n = 2 are valid solutions.
conclusion, any integers m and n will make [tex]\(6m^2 + 34n - 18\)[/tex]an even integer. One such solution is m = 1 and n = 2, and there are infinitely many other possible pairs of integers that satisfy this condition.
In the Numbers Game, a state lottery, four numbers are drawn with replacement from an urn containing balls numbered 0-9, inclusive. Find the probability that a ticket holder has the indicated winning ticket.
The first two digits in exact order.
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Thus, number of ways in which four numbers will be drawn from an urn containing balls numbered 0-9, inclusive is given by [tex]10^4=10,000[/tex]
The number of ways in which a ticket holder will have the first two digits in the correct order is the same as the number of ways in which two numbers will be drawn from the urn containing balls numbered 0-9, inclusive and is given by [tex]10^2=100[/tex]
Therefore, the probability that a ticket holder has the first two digits in exact order is given by [tex] \frac{100}{10,000} = \frac{1}{100} =0.01[/tex]
Final answer:
The probability of winning the indicated ticket in the Numbers Game is 1/100 or 0.01.
Explanation:
To find the probability of winning the indicated ticket in the Numbers Game, we will calculate the probability of the first two digits in exact order.
There are 10 numbers (0-9) to choose from in each digit, and since replacement is allowed, the probability of choosing the correct first number is 1/10. Similarly, the probability of choosing the correct second number is also 1/10.
Since the two selections are independent, we can multiply the probabilities together to find the probability of both selections being correct. Therefore, the probability of winning the indicated ticket is (1/10) * (1/10) = 1/100 or 0.01.
What is the GCF of 100xyz and 25xz
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Find the z-scores that bound the middle 74% of the area under the standard normal curve.
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The P values, the z-scores are approximately: z score (left) = -2.22 and z score (right) = 1.13.
We are given that;
The area middle= 74%
Now,
We need to find the z-scores that bound the middle 74% of the area under the standard normal curve.
Since 74% of the middle area is bounded, this means that there is 13% on the left side and another 13% on the right side.
P (left) = 0.13 and P (right) = 1 - 0.13 = 0.87. At these P values, the z-scores are approximately: z score (left) = -2.22 and z score (right) = 1.13
Therefore, by statistics the answer will be -2.22 and 1.13.
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The z-scores that bound the middle 74% of the area under the standard normal curve are approximately -1.04 and 1.04.
Explanation:To find the z-scores that bound the middle 74% of the area under the standard normal curve, we need to find the z-scores that bound 13% on each tail of the curve (100% - 74% = 26%, divided by 2). Using the z-table, we can find the z-scores for these tail areas:
The z-score for the lower tail area of 13% is approximately -1.04The z-score for the upper tail area of 13% is approximately 1.04Therefore, the z-scores that bound the middle 74% of the area under the standard normal curve are -1.04 and 1.04.
A pizza shop charges $9.50 for a pizza, plus $1.90 for each topping.
If the total charge for a pizza is dependent on the base price of the pizza plus the number of toppings ordered, what is the value of y at the y-intercept of the line representing this situation?
Answers
The value of y at the y-intercept for the given situation, representing the cost of a pizza without any toppings, is $9.50.
The value of y at the y-intercept represents the cost of a pizza without any toppings. When we graph the cost of the pizza as a function of the number of toppings, the y-intercept occurs when the number of toppings is zero (meaning no toppings were added).
Since the pizza shop charges $9.50 for a pizza without toppings, this is the value of y at the y-intercept. We can represent this relationship with a linear equation y = mx + b, where m is the slope (the cost per topping) and b is the y-intercept (the base price of the pizza). In this case, m is $1.90 and b is $9.50, so the equation would be y = 1.90x + 9.50. Therefore, the y-intercept is $9.50.
Deepak spent 178$ on food last month if he reduces by 15% the next month how much will he spend
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someone help mamd explain
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The simplest method:
Just multiply 3/4 by 500, to get 1500/4, or 375.
So 375 seats are filled.
A method that can be used with harder problems and will be more efficient in those cases:
Using proportions.
You could say,
?/500 = 3/4
And then cross multiply,
500 * 3 = 4 * x (say the ? = x)
1500 = 4x
/4 /4
x = 375
So either way we get the answer, 375 seats were filled.
Tran planned a rectangular pool and made a scale drawing using centimeters as the unit of measurement. He originally planned for the length of the pool to be 40 m but decided to change it to 32 m. If the length of the pool in his scale drawing is 8 cm, which statement about the change of scale is true?
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When Tran changes the actual length of the pool but keeps the scale drawing the same, the scale changes too. The original scale was 1 cm:5 m and the new scale is 1 cm:4 m.
Explanation:The question is about understanding the change in scale in a scale model. Tran's original scale was 8 cm (in the drawing) to 40 m (actual pool length), i.e., a ratio of 1 cm : 5 m. When he changes the length of the pool from 40 m to 32 m but keeps the drawing the same, the new scale becomes 1 cm: 4 m, since 8 cm (in the drawing) now corresponds to 32 m (actual pool length). Thus, the scale has been reduced proportionately.
Scale models and their measurements are an important concept in mathematics, particularly in understanding dimensions and ratios. In this scenario, the ratio of the drawing to the actual length of the pool changes when the dimensions of the pool change, but the drawing stays the same.
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What conclusion can be made based on this division problem?
24 ÷ 4 = 6
Twenty-four is 2 times greater than 6.
Twenty-four is 6 times greater than 4.
Six is 2 times greater than 4.
Six is 24 times greater than 4.
Answers
you can test it by multiplying 6 by 4
[tex]6 \times 4 = 24[/tex]
Answer:
Twenty-four is 6 times greater than 4.
Step-by-step explanation:
Here, the given expression,
[tex]24\div 4=6[/tex]
That is,
[tex]\frac{24}{4}=\frac{6}{1}[/tex]
By cross multiplication,
[tex]24=6\times 4[/tex]
⇒ 24 = 6 times of 4
Hence, by the given expression it is clear that twenty-four is 6 times greater than 4.
SECOND OPTION is correct.
James bought 12 cookies and ate 3 of them. He ate ____% of the cookies.
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So there are 9/12 cookies left.
So then John ate 3/12 cookies.
3/12 = 1/4
Convert into a decimal. .
1/4 = 0.25
0.25 = 25%
So, James bought 12 cookies and ate 3 of them. He ate 25% of the cookies.
I need help please fast
Answers
Answer:
16
Step-by-step explanation:
The number of divisors is the product of the powers of the prime factors after 1 has been added to each. In order for the total number of divisors to be 5, the number must be a prime raised to the 4th power. The only one in the given range is 2^4 = 16.
The divisors of 16 are 1, 2, 4, 8, 16.
_____
The smallest number with 5 prime factors is 2^5 = 32, so we assume you mean "divisors" when you say "factors."
WILL GIVE BRAINEST
if f(x)=2x-5 and g(x)=x2.....
Answers
[tex]f(x)=2x-5\\g(x)=x^2-4x-8[/tex]
Which means that
[tex](f+g)(x)=f(x)+g(x)=(2x-5)+(x^2-4x-8)[/tex]
And from there, it's just a matter of combining like terms to simplify.
The equation 9t = 27.99 models a constant rate situation. Drag and drop the value of 5t in the box. 5t = 3.11 15.55 18.99 36.99
Answers
Since in the equation 9t = 27.99 is basically multiply two numbers to get a product, divide 27.99 by 9:
27.99/9 = 3.11
3.11 is the value of t. So now that we've found the value of t, plug it into the equation 5t like so:
5*3.11 = 15.55
I hope this helped you out! :)
Then 5t would be 5(27.99 / 9).
How to find the inverse of h(x) = 3/ -x - 2 List steps.
Answers
First, you would have to switch the domain and range. In other words, swap x and y:
h(x) = 3/ -x-2
y = 3/ -x-2
x = 3/ -y-2
Then, when you have this, simply solve for y:
x = 3/-y-2
x (-y-2) = 3/-y-2 (-y-2)
x(-y-2) /x= 3 /x
-y-2 = 3/x
-y = 3/x + 2
y = -3/x -2
So the inverse would be
h⁻¹(x)= -3/x -2
a class donated $3.5 in nickels and dimes. In all there were 45 coins. how many were there of each
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So, 45 coins times the value of 5 cents would then be $2.25.
So that means we are 1.25 dollars short of 3.50 dollars.
If the dime is worth 5 cents more then the nickel is, we can just divide 1.25 by 5 cents to see how many extra 5 cents we need.
1.25/0.05 = 25
So therefore there were 25 dimes, and through that we can tell there were 20 nickels.
Latrell is packing boxes that can contain two types of items. Board games weigh 3 pounds and remote controlled cars weigh 1.5 pounds. The box can hold no more than 25 pounds.
Let x represent the number of board games. Let y represent the number of remote controlled cars.
Enter an inequality that represents the situation.
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hope this helps!
what is 150% of 128 ??
Answers
150/100 x 128
1.5 x 128
That will be 192
Hope I helped. Good luck
which expression is represented by the model below?
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Answer:
The answer is B 4x3
se an Addition or Subtraction Formula to simplify the equation. sin θ cos 3θ + cos θ sin 3θ = 0
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[tex]\bf sin(4\theta )=0\implies 4\theta =sin^{-1}(0)\implies 4\theta = \begin{cases} 0\\ \pi \\ 2\pi \end{cases}\\\\ -------------------------------\\\\ 4\theta =0\implies \measuredangle \theta =0\\\\ -------------------------------\\\\ 4\theta =\pi \implies \measuredangle \theta =\cfrac{\pi }{4}\\\\ -------------------------------\\\\ 4\theta =2\pi \implies \measuredangle \theta =\cfrac{2\pi }{4}\implies \measuredangle \theta =\cfrac{\pi }{2} [/tex]
Rewrite the following fractions as division problems. a. 7⁄11 b. 5⁄2 c. 9⁄10 d. 7⁄15
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1. For 7/11
Proceed as follows:
How many time will you find 11 in 7? This is not possible, so you will write down zero and put a decimal point in front of it. Then add zero at the back of 7 to get 70. How many time will 11 go in 70? That will be 6 times and you will have 4 remaining. Add zero to the back of the 4 to get 40. How many time will 11 go in 40? That will be 3 times and 7 will remain. You can continue to add zero to the remainder until you get as many decimal place as you desired.
From the calculations done above the answer we have obtained is 0.63.
Therefore, 7/11 = 0.63.
2. For 5/2, the same principle is applicable.
How many times will 2 go in 5? That will be 2 times remaining 1. Add zero to that one to obtain 10. How many times will 2 go in 10.That will be 5 times remaining zero.
Therefore 5/2 = 2.50.
3. For 9/10
How many times will 10 go in 9? That is not possible, so you will write down zero and put a decimal point in front of it. Add zero to 9 to obtain 90. How many times will 10 goes in 90. That will be 9 times exactly .
Therefore, 9/10 = 0.90.
4. For 7/15
How many time will 15 go in 7? That is not possible, so you write down zero with a decimal point in front of it. Add zero to 7 to get 70. How many times will 15 go in 70? That will be 4 times remaining 10. Add zero to the remaining 10 to obtain 100. How many time will 15 go in 100? That will be will be 6 times, remaining 10.
Therefore, 7/15 = 0.46
Final answer:
Fractions 7⁄11, 5⁄2, 9⁄10, and 7⁄15 can be rewritten as division problems by expressing each as the division of the numerator by the denominator.
Explanation:
To rewrite the given fractions as division problems, you simply need to express each fraction as a division of its numerator by its denominator. Here's how the fractions can be rewritten:
7⁄11 can be written as 7 divided by 11.
5⁄2 can be written as 5 divided by 2.
9⁄10 can be written as 9 divided by 10.
7⁄15 can be written as 7 divided by 15.
Each fraction represents a division problem where the numerator is the dividend and the denominator is the divisor.
A closed rectangular box has volume 32 cm3 . what are the lengths of the edges giving the minimum surface area?
Answers
The lengths of the edges of a rectangular box with a volume of 32 cm³ that would give the minimum surface area would all be equal, and that would be the cubic root of the volume, which is approximately 3.17 cm.
Explanation:The subject is Mathematics, specifically dealing with optimization of geometric shapes. This problem is related to the minimization of the surface area of a closed rectangular box given a fixed volume.
Given that the volume of a rectangular box, or a cuboid, is V = lwh where l is length, w is width, h is height. We know the volume is 32 cm³, we have the equation lwh = 32.
To minimize the surface area, we should make the box as close to a cube as possible because a cube has the smallest surface area for a given volume. So, to minimize the surface area, the lengths of the edges should all be equal, i.e., l = w = h.
Hence, the box that has the minimum surface area is a cube with edge length of the cubic root of the volume, which is ∛32 cm = approximately 3.17 cm.
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