Mathematics Middle School

A farmer has 300 ft of fencing with which to enclose a rectangular pen next to a barn. The barn itself will be used as one of the sides of the enclosed area.

What is the maximum area that can be enclosed by the fencing?

_______ft^2

Answers

Answer 1
This question can be answered in a number of ways depending on your level of mathematical sophistication (using calculus, graphing a parabola, and guess/check). I'll talk through all three in the order above.

They all start the same way: Let's call the length of the rectangle b and the width a. The perimeter of the rectangle (the distance all the way around -- that is, the length of the "fence") is 2a + 2b. Since the barn is used as one of the sides (let's say b) we can subtract b ... we don't need fencing for this side. That makes the perimeter 2a + b. Since we have 300 feet of fencing we set these equal: 2a + b = 300.

We can solve for b above by subtracting 2a from both sides of the equation so we get: b = 300 - 2a.

We are asked to maximize the area. The area of a rectangle is length times width. In this case, (a)(b) and since we just found an expression for b, we can substitute this to obtain: Area = [tex](a)(300-2a)=300a-2 a^{2} [/tex]. This is a parabola when graphed and we are asked to find where it's highest point occurs because this is where the area is greatest.

METHOD 1: Using Calculus
We are looking for the highest point of the equation [tex]Area = [tex](a)(300-2a)=300a-2 a^{2}[/tex]. Let's call the function R(a) since it is in terms of a. To find the maximum we take the first derivative, set it equal to zero and solve for a. As this is a parabola (with the leading coefficient being negative) it is concave down and the point we find will be the only maximum.

First we find the derivative of the function.
[tex]R(a)=(a)(300-2a)=300a-2 a^{2} [tex]

Next set it equal to 0 and solve for a.
R^{'}(a)=300-4a 0=300-4a -300=-4a 75 = a[/tex] [/tex]
This means that the maximum value (the largest area) occurs when a = 75.
At a = 75 the area is given by
What is the largest area? Let's us substitute 75 for a in the equation of the area. This gives: [tex]Area=a(300-2a)=(75)(300-(2)(75))=(75)(150)=11250[/tex] square feet.

METHOD 2: Graphing
We know that the area is given by. Area = [tex](a)(300-2a)=300a-2 a^{2} [/tex]. To make this more familiar let's use y for the area and x for the width (a). That means we have the equation: [tex](a)(300-2a)=300a-2 a^{2} [/tex].

This is a quadratic equation (the highest exponent of x is 2) and so the equation of a parabola (like the letter U or an upside down U). Since the leading coefficient is negative (-2) we know it opens downward (called concave down -- think of a frown). We are looking for the x-coordinate of the highest point called the vertex. We can find this by remembering that the x-coordinate of the vertex is given by [tex]x= \frac{-b}{2a} [/tex] where a is the leading coefficient the coefficient of the term with the x squared (here = -2) and a is the coefficient of the linear term (the one with x...here 300). That makes the x-coordinate of the vertex [tex]x= \frac{-300}{2(-2)}= \frac{-300}{-4}=75 [/tex].

The y-coordinate of the vertex is found by substituting 75 for x in the equation for the area. [tex]y=300x-2 x^{2} y=300(75)-2(75)^{2} [/tex]
[tex]y=22500- 2(5625) y=22500- 11250 y=11250[/tex]

Since we let x be the width of the rectangle. We know that the width is 75. The area of the rectangle we called y. So the maximum area is 11250 square feet which occurs when the width of the rectangle is 75.

You could have gotten to this point by generating a list of points for the parabola and graphing it carefully on graph paper (though it might not have been exact depending on how you scale your axis).

METHOD 3: Guess and Check
The area of the rectangle is given by Area = [tex](a)(300-2a)=300a-2 a^{2} [/tex]. Here you guess values for a, plug them into the formula to get the area and keep doing this until you are comfortable you have found the largest area. Not the best method but if it's a test and you forget everything else, it might get you close to an answer or even get you the exact answer (depending on how well you guess). It might take a while so I'd do this question last if this were a test.
 

Related Questions

If Jimmy increases a number by 10%, and then decreases this new value by 10%, what percent of the original number will the final result be?

Answers

If he increases the number by 10%, he will have 110% of the number. 
Then he reduces it by 10% of the new number, which s 11%. 
110% - 11% = 99%
The answer is 99%.
Hope this helps!

Matthew has saved a total of 10400 of which 2600 is invested in the stock market. What percent of his total saving is invested in the stock market

Answers

I think the answer is 25% because 2600 divided by 10400 equals 25%
Sorry if this is incorrect

Marci is going to use tile to cover her terrace. How much tile does she need?

Answers

Hi there:-}

The answer is 4*8+(16p/2)

32+8p

A=(32+8p)m^2

A=57.13 m^2 (to nearest hundredth of a m^2)

Hope this helps you>:)

~ TRUE BOSS

What does the y-intercept have to do with the graph?

Answers

It gives the other half of the coordinates.
Ex:(3, 6)
The y is the 6 the 6th number up the line.

The puzzles measures 12 inches by 12 inches.what is the area of the puzzle.

Answers

the answer is 144 because 12x12=144

200 grams of fertilizer are used for each tree in an orchard containing 1200 trees. at $2.75 per kilogram of fertilizer, how much does it cost to fertilize the orchard ?

Answers

To determine the cost to fertilize the trees, you need to figure out how many kilograms of fertilizer are needed. With the information given, you can determine The number of grams needed to fertilize 1200 trees. To do this you would multiply 200 times 1200. This equals 240,000 grams. To convert this to kilograms, divide 240000g by 1000 g. Every group of 1000 g is 1 kg. The answer is 240 kilograms. Multiply 240 kg by the price of $2.75 per kilogram to get $660 as the cost.

To fertilize an orchard of 1200 trees, with 200 grams of fertilizer per tree and the cost of fertilizer at $2.75 per kilogram, the total cost would be $660.

To calculate the cost to fertilize an orchard of 1200 trees, assuming 200 grams of fertilizer are used per tree, and the fertilizer costs $2.75 per kilogram, we need to follow these steps:

Convert the amount of fertilizer needed per tree into kilograms, as the cost is given per kilogram. Since there are 1000 grams in a kilogram, 200 grams is equal to 0.2 kilograms.

Multiply the amount of fertilizer needed per tree by the number of trees to find the total amount of fertilizer needed for the orchard. This is 0.2 kg/tree times 1200 trees = 240 kilograms.

Multiply the total kilograms of fertilizer by the cost per kilogram to find the total cost of fertilizing the orchard. This is 240 kg times $2.75/kg = $660.

Therefore, the total cost to fertilize the orchard is $660.

A machinist creates a washer by drilling a hole through the center of a circular piece of metal. If the piece of metal has a radius of x + 7 and the hole has a radius of x + 6, what is the area of the washer?

Answers

The area of the washer (which is a circle, but the inner part cut out) is given by:
[tex]A_c = \pi r^2[/tex]

Now, we need to find the radius of the entire metal circle which the hole was drilled on:
[tex]R_w = (x+7) - (x+6) = 1[/tex]

Plug it in to find the area of the metal plate:
[tex]A_p = \pi (1)^2 = \pi [/tex]

Now, you subtract the area of the hole from the area of the plate. The area of the hole will be:
[tex]A_h = \pi (x+6)^2[/tex]
[tex]A_h = \pi (x^2+12x+36) =  [/tex]

Subtract the area of the hole from the metal plate to get:
[tex]A_w = \pi - (\pi x^2+12 \pi x+36 \pi) = \pi x^2+12 \pi x -35 \pi[/tex]

That's your answer, although it looks wrong because not much information was provided. 

The hypotenuse of a right triangle is 37 mm. One leg of the right triangle is 12 mm. What is the length of the other leg? 30 mm 35 mm 40 mm 47 mm


The legs of a right triangle are 10 cm and 24 cm. What is the length of the hypotenuse?

 22 cm 26 cm 30 cm 34 cm

Answers

leg=35 hypotenuse=26

Principle = $15,500 Interest Rate = 8.75% Time = ? Years Simple Interest = $5425.00

Solve for the Time in Years

Answers

We must find time by using the simple interest formula:

I = P x i x t

I is the interest
P is the total principal 
i is the rate of interest per year
t is the total time in years

In this example, when we plug our known values in, our formula looks like this: 

t = [tex] \frac{I}{i*p} [/tex]

t = [tex] \frac{5425}{0.0875 * 15500} [/tex]

t = 4

So time (in years) would be 4 years. 

Landons front lawn measures 20 feet by 8 feet he counts to find that there are 120 blades of grass per square inch how many blade do grass are in the entire lawn use proportions

Answers

First step is to convert feet into inches:
1 foot = 12 inches
20feet = 20*12=240 inches
8feet = 8*12=96 inches

Now we can calculate total area:
A = height * width
A= 240 * 96
A=23040 inches²

We know the amount per one inch² and we need to find out total number of blades:

1 inch²                          120 blades
23040 inches²               x blades

Now we cross-multiply numbers:
1*x = 23040 * 120
x= 2764800

There are total of 2764800 blades in the entire lawn

An investment of $900 is invested at a rate of 4%. The function that models this situation if f(t)=900(1+0.4^t) where does x represent time in years what is the rate of time in years. What is the rate of change of interval (3,10)

Answers

Final answer:

To find the rate of change of the interval (3,10) for the given function, you need to calculate the difference in the function values at those two time points and divide it by the difference in time. The rate of change of the interval (3,10) is: ((900(1 + 0.4^10)) - (900(1 + 0.4^3))) / (10 - 3)

Explanation:

The given function f(t) = 900(1 + 0.4^t) models an investment of $900 at a rate of 4% over time, where t represents time in years. To find the rate of change of the interval (3,10), you need to find the difference in the function values at those two time points and divide it by the difference in time.

So, the rate of change of the interval (3,10) is:

((900(1 + 0.4^10)) - (900(1 + 0.4^3))) / (10 - 3)

The table shows the change in the low temperature in Albany compared to the day before.
Wednesday : +3.5
Thursday: -4.1
Friday : 0
Saturday: -2.2
On which day did the low temperature change the most?

Answers

Answer: On Thursday the low temperature change the most.

Explanation :

From the given table we can conclude that change is the lowest in Thursday .

And there is no matter if it is positive or negative

Because change defines final - initial only i.e. difference between final and initial

So, we consider absolute value only.

According to which Thursday has the atmost change i.e 4.1 which is the lowest amongst the other .

So ,on Thursday the low temperature change the most.

The day with the most temperature change in Albany compared to the day before is Thursday.

The day with the most temperature change was Wednesday.

To determine the most significant change, we calculate the absolute difference for each day's temperature change:

Wednesday: |+3.5| = 3.5Thursday: |-4.1| = 4.1Friday: |0| = 0Saturday: |-2.2| = 2.2

Therefore, Thursday had the most substantial temperature change.

PLEASE ANSWER FAST!!!!

A rectangular prism has a length of 114 centimeters, a width of 4 centimeters, and a height of 314 centimeters. What is the volume of the prism? Enter your answer in the box as a simplified mixed number or a decimal.

Answers

The answer is 143184!

9. 10. 11. And need work or won’t get credit

Answers

Answer:

9) 22 - 5(6v - 1) = -63

22 - 30v + 5 = -63

27 - 30v = -63

-30v = -90

v = 3

10) 18x - (8x - 7) = 67

18x - 8x + 7 = 67

10x = 60

x = 7

11) 8(-2x - 4) + 12 = -52

8(-2x - 4) = -64

-2x - 4 = -8

-2x = -4

x = 2

y=-x+7 , y=x-1 and show your work Help !

Answers

This is a system of equations. The best way to solve it is by realizing that you have two equations equal to y. Since y = y, it follows that [tex]-x+7=x-1[/tex]. Solving for x, you get 4.

[tex]7=2x-1 \\ 8=2x \\ 4=x[/tex]

Then, substitute 4 into one of the original equations to find y.

[tex]y=(4)-1=3[/tex].

The solution is [tex](4,3)[/tex]. I have attached a graphical representation of the solution as well.

What is the same as 98.52 divided by 10

Answers

Dividing 98.52 by 10 involves moving the decimal one place to the left, resulting in 9.852. This process is the same for all decimals when dividing by powers of 10, such as 10, 100, or 1000.

When we divide a number by powers of 10, we're effectively moving the decimal point to the left by as many places as there are zeros in the power of ten. For instance, dividing 1.9436 by 10 moves the decimal one place to the left, becoming 0.19436. Similarly, dividing 98.52 by 10 moves the decimal one place to the left as well, giving us the simplified answer of 9.852.

If we were to multiply a number by powers of 10, we would do the opposite, moving the decimal to the right. For example, to multiply 45.86 by 100, you would move the decimal two places to the right to get 4586.

In conclusion, 98.52 divided by 10 is 9.852. Even if the answer is rounded, the process remains the same; you simply adjust the decimal point accordingly based on the power of 10 you are dividing by.

I believe he area is incorrect. So can somebody help me to figure out the area and arc length. PLEASE show your work. So I can understand :)

Answers

To find the area of a sector, you must use the formula: [tex]a= \frac{central angle}{360} * \pi r ^{2} [/tex], so, since we know the degree measure, which is 80, and the radius, 1.3, we plug it into the formula and get: 1.17984257435

To find the arc length, you must use the formula: [tex] \frac{arc degree}{360} * 2 \pi r[/tex], then we plug in 80 as our arc degree and 1.3 as our radius, and we get: 1.81514242207

The sum of two numbers is -1. When twice the first number and four times the second number are added, it equals -10. What are the two numbers?

Answers

let the two number be x and y.
x+y=-1......i
2x+4y=-10.....ii
solving the simultaneous equation we have:
from equation i, making x the subject we get:
x=-1-y........iii
substituting iii in ii we get:
2(-1-y)+4y=-10
-2-2y+4y=-10
2y=-10+2
2y=-8
y=-8/2=-4
substituting the value of y in iii we get:
x=-1-(-4)
x=-1+4
x=3
therefore the first number is 3 and the second number is -4


You are a farmer and want to spend under $35,000 on farm equipment. You need a hay bailer that costs $6,250 and several plowing disks cost $2,500 each. Write an inequality that models how many plowing disks could be purchased within your budget. What is the maximum number of plowing disks you can buy?

Answers

Inequality: $6,250+$2,500p > $35,000

Maximum plowing disks: 10 which would cost you $31, 250

Answer:

A) 2,500x + 6,250 < 35,000; 11 plowing disks

Step-by-step explanation:

shannon uses 30 inches of ribbon to make one bow.how many feet are needed to make 10 bows?

Answers

30*10=300/12=25 ft hope i helped! 
she needs nearly 3 feet of ribbon
hope this helps

Kade makes 5/6 cup salad dressing. He uses 2/3 cup dressing on a salad. How much salad dressing is left

Answers

To solve [tex] \frac{5}{6} - \frac{2}{3} [/tex], we need to find a common denominator. If we multiply [tex] \frac{2}{3} [/tex] by [tex] \frac{2}{2} [/tex], we get [tex] \frac{4}{6} [/tex]. The reason we can do this is because two over two is really just equivalent to 1, and multiplying a number times one equals the original number. So if you're subtracting  [tex] \frac{4}{6} [/tex] from [tex] \frac{5}{6} [/tex], you have


[tex] \dfrac{5}{6} - \dfrac{4}{6} = \dfrac{5-4}{6} = \dfrac{1}{6} \ cups[/tex].


Kade has 1/6 cup of salad dressing left after using some on a salad. We found this by converting 2/3 cup to 4/6 cup and then subtracting it from the 5/6 cup he initially had.

To find out how much salad dressing Kade has left after using some on a salad, we need to subtract the amount used from the total amount he made. Kade made 5/6 cup of salad dressing and used 2/3 cup on a salad. To perform the subtraction, both fractions need a common denominator. The smallest common denominator for 6 and 3 is 6. The fraction 2/3 can be converted to 4/6 so that both fractions have the same denominator. Now we can subtract:

Convert 2/3 cup to 4/6 cup (Since 3*2 = 6, and 2*2 = 4).Subtract 4/6 cup from 5/6 cup: 5/6 - 4/6 = 1/6 cup.

Hence, Kade has 1/6 cup of salad dressing left after using it on the salad.

for every pound a company spends on advertising it spends £0.64 on its websites. express the amount spent on advertising to the amount on its website it's website as a ratio it's in its simplest form

Answers

The ratio expression is £ 1.00 : £0.64 =  100:64
                                       = 25:16

Solution:

we are given that

For every pound a company spends on advertising it spends £0.64 on its websites.

we have been asked to express the amount spent on advertising to the amount on its website it's website as a ratio it's in its simplest form.

We can find the required ratio as below

[tex]Ratio=\frac{1}{0.64} \\ \\ Ratio=\frac{100}{64} \\ \\ Ratio=\frac{25}{16} \\[/tex]

Hence the required ratio is 25:16.

Let ∠1, ∠2, and ∠3 have the following relationships.
∠1 and ∠2 are acute vertical angles.
∠3 is an obtuse angle adjacent to both ∠1 and ∠2.
What is the sum of the measure of ∠1 and the measure of ∠3?

Answers

3 can be anything greater than (but not equal to) 90 and 1 can be anything less than (but not equal to) 90 as long as angle 3 + angle 1 = 180 degrees.

"The value of [tex]\angle1 +\angle3[/tex] lies between [tex]90^{\circ} &lt; \angle1+\angle3&lt; 180^{\circ}[/tex].

Given,  

[tex]\angle 1,\angle2, \angle3[/tex] have the following relationships.

[tex]\angle 1[/tex]  and[tex]\angle2[/tex] are acute vertical angles. [tex]\angle 3[/tex] is an obtuse angle adjacent to both [tex]\angle 1[/tex] and [tex]\angle 2[/tex].

We have to find the sum of measure of [tex]\angle1[/tex] and the measure of [tex]\angle3[/tex].

Angle sum property:

We know that the angle sum property of triangle states that the sum of all angles of a triangle is equal to [tex]180 ^{\circ}[/tex].

[tex]\angle 1+\angle 2+\angle3= 180^ {\circ}[/tex]

[tex]\angle1+\angle3= 180-\angle2[/tex]

So, the value [tex]\angle1 and \angle3[/tex] is always greater than [tex]90{^\circ}[/tex] because [tex]\angle 3[/tex] is the obtuse angle and it is being added to [tex]\angle1[/tex],

so it is always greater than right angle and less than [tex]180^{\circ}[/tex].

Hence the value of [tex]\angle1 +\angle3[/tex] lies between [tex]90^{\circ} &lt; \angle1+\angle3&lt; 180^{\circ}[/tex].

For more details on angle sum property follow the link:

https://brainly.com/question/4316040

A ________ is a monomial or the sum or difference of two or more monomials. Base, polynomial, linear term,quadratic equation

Answers

The answer is:

 "A polynomial is a monomial or the sum or difference of two or more monomials".

 By definition, the terms of a Polynomial Function have the form ax^n, where "n" is a natural number. 

 To classify a polynomial, you must count the number of its terms. For example:

 5x²-2x  (This is a binomial, because it is formed by two terms).
 x³-3x+1 (This is a trinomial, because it has three terms):

 To identify the degree of a polynomial, you must see the term which has the highest exponent. For example, the degree of following polynomial is 2.

 4x²-x+3 

 The degree of this polynomial is 3:

 x³+2x²-x+10

Number 14
I need help with this

Answers

n is five more than twelve should be n=12+5 the example out the 5 in the wrong side of the equation

PLZ HELP I BEG U IT MAKE NO SENSE

Answers

Segrigation is excluding a group or people in a place or having them moved. So when they say segrigation on buses they are talking about forcing black people to stand in the back or sit and get up when a white person needed a seat.
The correct answer is the SECOND CHOICE : The court said that bus companies could not tell passengers where to sit, based on the color of their skin.

Help me with this one question and think closely it is not 8 or 800 question 36

Answers

(1 L)/(100 ft^2) * (800 m^2) * (0.26417205236 gal/L) * ((1 ft)/(0.3048 m))^2
≈ 22.75 gal

___
The conversion from liters to gallons is accurate to the digits shown, but is not exact.

you are asked to determine if the two events "a randomly selected student has asthma" and "a randomly selected student has a household member who smokes" are independent. what probabilities could you calculate to answer this question?

Answers

Asthma is more likely to occur in someone who lives in a household with at least one person who smokes
Let's define the two events A and B to be...
event A = student has asthma
event B = at least one household member smokes

To find the probability of event A happening, we count the total number of people who have asthma out of the total in the survey. So we simply look at the "student has asthma" row and look at the "total" column. This value is 182. There are 182 people who have asthma. This is out of 937 (bottom right corner).

Divide the values: 182/937 = 0.194237
This value is approximate

So P(A) = 0.194237 approximately
The P means probability
P(A) = probability of event A happening

----------------------------------

Now compute P(B)
P(B) = 395/937
P(B) = 0.421558
Notice I'm dividing the total value in the "at least one member smokes" column over the grand total of 937

----------------------------------

Next we compute P(A and B) which is the probability both events occur at the same time. 

P(A and B) = 113/937
P(A and B) = 0.120598
The 113 is the amount of people who have asthma AND have at least one household member that smokes (row1, column2)

----------------------------------

If events A and B were independent, then P(A)*P(B) should be equal to P(A and B)

Let's see if that is the case
P(A and B) = P(A)*P(B)
0.120598 = P(A)*P(B)
0.120598 = 0.194237*0.421558
0.120598 = 0.08188216
we get two different results so P(A and B) = P(A)*P(B) is not true

Therefore, A and B are not independent events. They are said to be dependent events.

using similarity solve for x. show all work

Answers

To find x, let the altitude=h
First find h.
h²= 9 x 16
h²=144
Take square root.
h=12
Now to find x, we use Pythagorean theorem.
c²=a²+b²
Put values
x²=9²+12²
x²=81+144
x²=225
Take square root.
x=15

Answer: x=15

graphing system y=2x-2 2y=4x- 4

Answers

These two equations describe the same line.

___
In the graph here, the graph of one equation is shown as a dotted line, not be cause those are the only points on the graph, but because I wanted you to see the other equation describes exactly the same line.

Answer:

These two equations describe the same line.

___

In the graph here, the graph of one equation is shown as a dotted line, not be cause those are the only points on the graph, but because I wanted you to see the other equation describes exactly the same line.

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