Mathematics College

There are $400 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $15 perfoot, and the fencing for the other three sides costs $5 per foot. Consider the problem of finding the dimensions of the largest possible garden.(a) what would be the objective equation and what would be the constraint equations?(b) Express the quantity to be maximized as a function of x.(c) Find the optimal values of x and y.

Answers

Answer 1

Answer:

The answer to this question can be described as follows:

Step-by-step explanation:

The goal is to increase the area and therefore

Target formula: A = xy

The limited amount of money, which we have for the fence, and it only has $400

The equation of constraints: 400=(fence costs parallel to roads) +( 3 other side cost)

$\to 400 = 15x+ 5(5y+x)\\\\ \to400 =15x+25y+5x\\\\ \to 400=20x+25y\\$

To solve y we use the limiting equation:

$\to 400=20x+25y\\\\ \to 25y= 400-20x\\\\\to y=\frac{(400-20x))}{25}\\\\\to y=16-\frac{4}{5}x\\\\$

Now, put the value in the y and maximise as much as possible into our goal equation:

A function for:

$\to A(x)= x(16-(\frac{4}{5}x)\\\\\to A(x) =16x-\frac{4x^2}{5}$

Select the critical value the A(x):

$\to \frac{dA}{dx} 16-\frac{8x}{5}=0 \\\\\to 16=\frac{8x}{5}\\\\\to 16 \times 5 =8x\\\\\to 80=8x \\\\\to x=\frac{80}{8}\\\\\to x=10\\\\$

We now need, with the second derivative, to ensure that this value is maximum:

$\to \frac{d^2A}{dx^2} = \frac{-8}{5}\\\\ \to \frac{d^2A}{dx^2}(10) < O$

not only is there a relative maximum at x = 10, but we can conclude that the

maximum occurs as$x = 10 \ since \ \frac{d^2A}{dx^2}< O$ for all x. We also need y:

$\to y= 16-\frac{4}{5}x\\\\\to y=16-\frac{4}{5}*10\\\\\to y= 16-8\\\\\to y=8\\\\$

Related Questions

Ronnie made a scale drawing of a shopping center. A bakery in the shopping center is 7 inches wide in the drawing. The actual bakery is 42 feet wide. What is the scale of the drawing?

Answers

Answer:

1:72

Step-by-step explanation:

The scale of a drawing is the ratio of the length in the drawing to the real length.

Drawing length: 7 inches

Real length: 42 ft × 12 in. / ft = 504 in.

Scale:

7 in. to 504 in.

Divide both numbers by 7:

1 in. to 72 in.

Scale: 1:72

Which equation represents the distance (d) a car travels, in time (t), when driving 65 miles per hour? How far did the car travel in 3 hours?

Answers

Answer:

195 miles

Step-by-step explanation

D = 65 x 3

Answer:

D=65*tthe car traveled 195 miles in 3 hours

Step-by-step explanation:

D=65*t

where t is time in hours and D is the total distance traveled in miles

In 3 hours, the car will travel:

D=65*3

D=195

So, the car traveled 195 miles in 3 hours

an inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec. Find the rate of change of the water depth when the water depth is 10 ft.

Answers

Answer:

the rate of change of the water depth when the water depth is 10 ft is; $\mathbf{\dfrac{dh}{dt} = \dfrac{-25}{100 \pi} \ \ ft/s}$

Step-by-step explanation:

Given that:

the inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.

We are meant to find the rate of change of the water depth when the water depth is 10 ft.

The diagrammatic expression below clearly interprets the question.

From the image below, assuming h = the depth of the tank at a time t and r = radius of the cone shaped at a time t

Then the similar triangles ΔOCD and ΔOAB is as follows:

$\dfrac{h}{r}= \dfrac{20}{8}$ ( similar triangle property)

$\dfrac{h}{r}= \dfrac{5}{2}$

$\dfrac{h}{r}= 2.5$

h = 2.5r

$r = \dfrac{h}{2.5}$

The volume of the water in the tank is represented by the equation:

$V = \dfrac{1}{3} \pi r^2 h$

$V = \dfrac{1}{3} \pi (\dfrac{h^2}{6.25}) h$

$V = \dfrac{1}{18.75} \pi \ h^3$

The rate of change of the water depth is :

$\dfrac{dv}{dt}= \dfrac{\pi r^2}{6.25}\ \dfrac{dh}{dt}$

Since the water is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec

Then,

$\dfrac{dv}{dt}= - 4 \ ft^3/sec$

Therefore,

$-4 = \dfrac{\pi r^2}{6.25}\ \dfrac{dh}{dt}$

the rate of change of the water at depth h = 10 ft is:

$-4 = \dfrac{ 100 \ \pi }{6.25}\ \dfrac{dh}{dt}$

$100 \pi \dfrac{dh}{dt} = -4 \times 6.25$

$100 \pi \dfrac{dh}{dt} = -25$

$\dfrac{dh}{dt} = \dfrac{-25}{100 \pi}$

Thus, the rate of change of the water depth when the water depth is 10 ft is; $\mathtt{\dfrac{dh}{dt} = \dfrac{-25}{100 \pi} \ \ ft/s}$

an inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole

Answer:

the rate of change of the water depth when the water depth is 10 ft is;

Step-by-step explanation:

Given that:

the inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.

We are meant to find the rate of change of the water depth when the water depth is 10 ft.

The diagrammatic expression below clearly interprets the question.

From the image below, assuming h = the depth of the tank at a time t and r = radius of the cone shaped at a time t

Then the similar triangles ΔOCD and ΔOAB is as follows:

( similar triangle property)

h = 2.5r

The volume of the water in the tank is represented by the equation:

The rate of change of the water depth is :

Since the water is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec

Then,

Therefore,

the rate of change of the water at depth h = 10 ft is:

Thus, the rate of change of the water depth when the water depth is 10 ft is;

Step-by-step explanation:

When writing a proof, how do you construct the first statement?
A) By writing the justification for the first statement in the right column.
B) By copying the “prove” statement(s) from the original problem.
C) By writing the next logical statement from the current one.
D) By copying the “given” statement(s) from the original problem.

Answers

When writing a proof, you should construct the first statement by copying the “prove” statement(s) from the original problem. The Option B is correct.

How should you construct the first statement in a proof?

When constructing the first statement in a proof, it is important to begin with copying the “prove” statement(s) from the original problem. This involves writing the next statement based on the given or previously proven statements.

It is not helpful to write a justification for the first statement in the right column without considering its logical connection to the problem. By beginning with a logically connected statement, the proof can proceed in a clear and organized manner which leads to a valid conclusion.

Answers

The median of the distances is M = 6.5 kilometers

Given data ,

To find the median of a set of numbers, we arrange the numbers in ascending order and then locate the middle value. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the average of the two middle numbers.

Arranging the given set of numbers in ascending order: {4, 5, 8, 11}

Since the set has an even number of values, the median is the average of the two middle numbers. In this case, the two middle numbers are 5 and 8. To find the average, we add these two numbers and divide by 2:

(5 + 8) / 2 = 13 / 2 = 6.5

Hence , the median of the given set {4, 5, 8, 11} is 6.5

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Translate this sentence into an equation.
44 is the sum of 19 and Diane's age.
Use the variable d to represent Diane's age.

Answers

Answer:

19 + d = 44

dianas age is 25 :)

Step-by-step explanation:

the sum of = addition ! hope this helps !

Write an equation of the locus of all points equidistant from the x axis and the y axis and whose coordinates have the opposite signs.

Answers

Answer:

y=-x

Step-by-step explanation:

Since the y-axis and x-axis are intersecting lines then you need to apply the locus of intersecting lines which would be the angle bisector and since the problem is asking for the opposite signs you need to make it -x instead of x.

Hope this helps

Calls to a customer service center last on average 1.4 minutes with a standard deviation of 1.7 minutes. An operator in the call center is required to answer 52 calls each day. Assume the call times are independent. What is the expected total amount of time in minutes the operator will spend on the calls each day

Answers

Answer:

the expected total amount of time in minutes is 88.4

Step-by-step explanation:

The computation of the expected total amount of time in minutes is given below:

= Standard deviation × needed to answer calls each day

= 1.7 × 52

= 88.4

hence, the expected total amount of time in minutes is 88.4

Ava collects 9 insects in her net issac collects 8 more then Ava how many insects does Isaac collect

Answers

Answer:

Seventeen insects

Step-by-step explanation:

9+8=17

Answer:

Step-by-step explanation:

available has 9, issac gets 8. 9 plus 8 equals 17

Write and solve an equation to find the value of x.

Answers

The value of x for each item is given as follows:

28. x = 5.

29. x = 3.44.

How to obtain the value of x in each item?

For item 28, we apply the crossing chord theorem, which states that the products of the parts of the chords are equal, hence the value of x is obtained as follows:

16x = 10 x 8

16x = 80

x = 5.

For item 29, we apply the two secant theorem, hence the value of x is obtained as follows:

10(x + 10) = 12(12 + 25)

10x + 100 = 444

10x = 344

x = 3.44.

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What is the decimal equivalent of 1/18

Answers

The decimal equivalent of 1/18 is 0.125

0.125
Hope this helps!!!

At the start of a game of marbles, Peter and Jack had 160 marbles in all. In the first round, Peter lost 3/5 of his marbles to Jack. In the second round, James lost 3/7 of his marbles to Peter. At the end of the second round of the game, they had the same number of marbles. How many marbles did each of them have at first?

Answers

Answer: Therefore, at the start of the game, Peter had 80 marbles and Jack had 80 marbles.

Step-by-step explanation:

Three-year-old boys in the United States have a mean height of 38 inches and a standard deviation of 2 inches, and 10-year-old girls have a mean height of 54.5 inches and a standard deviation of 2.5 inches.
Mrs. Davis has two children: a 3-year-old boy 43 inches tall and a 10-year-old girl 57 inches tall.
Which of Mrs. Davis' children is more unusually tall for his
or her age and gender? Explain, showing any calculations you perform.

Answers

The 3-year-old boy is more unusually tall for his age and gender than the 10-year-old girl.

To solve this problem

A z-score calculates how many standard deviations an observation is from the mean. An observation is said to be above the mean if the z-score is positive, while it is said to be below the mean if the z-score is negative.

For the 3-year-old boy, the z-score is:

z = (43 - 38) / 2 = 2.5

The z-score for the 10-year-old girl is:

z = (57 - 54.5) / 2.5 = 1

The 3-year-old boy has a higher z-score than the 10-year-old girl, indicating that he is more unusually tall for his age and gender. This is due to his height, which is higher than the average for his age group 38 inches .

Therefore, the 3-year-old boy is more unusually tall for his age and gender than the 10-year-old girl.

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Which triangle is a 30-60-90 triangle

Answers

Answer:

I think it's A on edge.

Step-by-step explanation:

Find the area of a quarter circle with the radius of 6cm

Answers

Answer:

28.27cm³

Step-by-step explanation:

Area of a circle:

$A=\pi r^2$

First, find the area of a full circle with this given radius:

$\rightarrow A=(3.14159)(6)^2\\\rightarrow A=(3.14159)(36)\\\rightarrow A=113.09734\\$

Now, to find the radius of a quarter circle with the same radius, just divide by 4:

$\rightarrow A = \frac{113.09734}{4}\\\rightarrow A = 28.27433$

The radius of the quarter circle is about 28.27cm³

find the area of the circle which is circumscribed about the right triangle with legs 6 and 8

Answers

The area of the circle circumscribed about the right triangle with legs 6 and 8 is approximately 78.54 square units.

To find the area of the circle circumscribed about a right triangle, we can use the fact that the diameter of the circle is equal to the hypotenuse of the right triangle. In this case, the legs of the right triangle are given as 6 and 8.

Using the Pythagorean theorem, we can find the length of the hypotenuse:

$c^2 = a^2 + b^2$

where c is the length of the hypotenuse, and a and b are the lengths of the legs.

Substituting the values:

$c^2 = 6^2 + 8^2$

$c^2 = 36 + 64\\ c^2 = 100$

Taking the square root of both sides:

$c = \sqrt{100} \\ c = 10$

Therefore, the length of the hypotenuse is 10 units.

Since the diameter of the circumscribed circle is equal to the hypotenuse, the radius of the circle is half the length of the hypotenuse, which is 10/2 = 5 units.

Now we can calculate the area of the circle using the formula:

Area = π $r^2$

where r is the radius of the circle.

Area = π $5^2$

Area = π × 25

Area = 78.54 square units

Hence, the area of the circle circumscribed about the right triangle with legs 6 and 8 is approximately 78.54 square units.

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The regular price of a television is $250. A discount offered on the television is 20% of the regular price. What is the discount amount?

Answers

Answer:

The discounted TV would cost $200 because the 20% discount takes off $50

4/5 times 1/5 PLZ ANSER I NEED HELP

Answers

Answer:

.16 or 4/25

Step-by-step explanation:

Answer:

4/25

Step-by-step explanation:

$\frac{4}{5}$×$\frac{1}{5}$

multiply tops together and denominators together and get $\frac{4}{25}$

Adam’s credit card calculates finance charges using the adjusted balance method and a 30-day billing cycle. The table below shows his use of that credit card over three months.

Date
Amount ($)
Transaction
4/1
626.45
Beginning balance
4/10
37.41
Purchase
4/12
44.50
Purchase
5/3
65.50
Payment
5/16
24.89
Purchase
5/20
104.77
Payment
6/6
23.60
Payment
6/10
15.00
Purchase
6/14
51.85
Purchase

If Adam’s credit card has an APR of 14.63%, what is Adam’s balance at the end of June?
a.
$629.42
b.
$629.66
c.
$627.27
d.
$628.40

Answers

Adam's balance at the end of June is $627.27

The adjusting balance method is used to determine the interest that would be paid by a credit card owner. The interest that would be paid is determined at the end of a period after all transactions have been adjusted for.

For example, if I have $100 in my credit card. If I buy a shoe worth $50 and deposit $20. My balance at the end of the month would be ($100 + $50 - $20) = $130.

Adam's balance can be determined by adding the amount he spent on purchases to the beginning balance and subtracting the payments he made.

$626.45 + $37.41 + $44.50 - $65.5 +$24.89 - $104.77 +$23.60 + $15 + $51.85 = $605.23

The balance at the end of the month would be the sum of the interest and the amount in his balance

Interest earned = balance x interest rate

Interest rate = 14.63% / 4 = 3.66%

Interest earned = $605.23 x 0.0366 = $22.14

Balance = $22.14 + $605.23 = $627.37

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Answer:

Step-by-step explanation:

627.27. Just had it on my test

find the sum of the interior angles of a regular octagon

Answers

we know that

the sum of the interior angles in any polygon is equal to

S=180(n-2)

where

n is the number of sides

In this problem

n=8

substitute

S=180(8-2)

S=180(6)=1,080 degrees

therefore

the answer is 1,080 degrees

Use the following table to find the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.
Students on the Student Government Board
On-Campus Housing Off-Campus Housing
Freshman 2 2
Sophomore 2 4
Junior 0 3
Senior 4 2
Graduate Student 2 0

Answers

The probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 8/25 or 0.32 (rounded to the nearest millionth).

1. Calculate the total number of students on the Student Government Board by summing up the numbers in the table:

Total Students = 2 + 2 + 2 + 4 + 0 + 3 + 4 + 2 = 19

2. Calculate the total number of graduate students on the Student Government Board:

Total Graduate Students = 2 + 0 = 2

3. Calculate the total number of students living in on-campus housing:

Total On-Campus Housing = 2 + 2 + 0 + 4 + 2 = 10

4. Calculate the probability of selecting a graduate student from the Student Government Board by dividing the total number of graduate students by the total number of students:

Probability of Graduate Student = Total Graduate Students / Total Students = 2 / 19

5. Calculate the probability of selecting a student living in on-campus housing by dividing the total number of students in on-campus housing by the total number of students:

Probability of On-Campus Housing = Total On-Campus Housing / Total Students = 10 / 19

6. Calculate the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing by summing up the probabilities from steps 4 and 5:

Probability = Probability of Graduate Student + Probability of On-Campus Housing = 2 / 19 + 10 / 19

7. Simplify the fraction if necessary. In this case, the fraction cannot be simplified further, so the final probability is 2 / 19 + 10 / 19 = 12 / 19.

8. Convert the fraction to a decimal by dividing the numerator by the denominator: 12 / 19 ≈ 0.631578947, which rounds to 0.632 (rounded to the nearest thousandth).

9. Finally, express the probability as a fraction in lowest terms: 12 / 19 is already in lowest terms.

Therefore, the probability that a randomly chosen member of the Student Government Board is a graduate student or lives in on-campus housing is 12/19 or approximately 0.632 (rounded to the nearest thousandth).

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The basic rate pay is K8.20. If the overtime is paid at time-and-a-quarter, what is the overtime rate of pay?

Answers

The overtime rate of pay is K10.25.

The overtime rate of pay can be calculated by multiplying the basic rate pay by the time-and-a-quarter factor. In this case, the basic rate pay is K8.20.

To determine the overtime rate of pay, we need to calculate one-quarter (1/4) of the basic rate pay, and then add that amount to the basic rate pay. One-quarter of K8.20 is calculated as (1/4) * K8.20 = K2.05.

By adding the calculated overtime amount to the basic rate pay, we get the overtime rate of pay: K8.20 + K2.05 = K10.25.

Therefore, the overtime rate of pay is K10.25.

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solve log2+log16/15+12log25/24+7log8
$ log_{10}2+ log_{10} \frac{16}{15} + 12 log_{10} \frac{25}{24} + 7 log_{10} \frac{81}{80 = 1$
1/80=1

Answers

Answer:

the answer is -non existent

Step-by-step explanation:

there is no answer to this problem

Jasmine has a circular swimming pool with a radius of 4.2 meters. What is the circumference of the pool? Use 3.14 for π
. Round to the nearest hundredth if necessary.
__ m

Answers

If the radius of Jasmine's swimming pool is 4.2 meter, then it's circumference is 26.4 meters.

The "Circumference" of a circle is known as the distance around the boundary of a circle.

The circumference of a circle is given by the formula : 2 × π × radius,

where π (pi) is a mathematical constant approximately equal to 3.14,

We are given that Jasmine's swimming pool has a radius of 4.2 meters.
So, we can calculate the circumference of the pool as :

⇒ Circumference = 2 × 3.14 × 4.2 meters,

⇒ Circumference ≈ 26.4 meters,

Therefore, the circumference of Jasmine's swimming pool is 26.4 meters.

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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Answers

Answer: Why is the question blank

Also the answer is 5.2

Step-by-step explanation:

How many solutions does the following equation have? -14(z-5)=-14x+70

Answers

Answer:

Infinite amount of solutions

Step-by-step explanation:

Parallel lines have no solution

Same lines have infinite solutions

Intersecting lines have 1 solution

Step 1: Write out equation

-14(x - 5) = -14x + 70

Step 2: Distribute -14

-14x + 70 = -14x + 70

Here we see that we have 2 exact same lines. Therefore, we have infinite amount of solutions.

Alternatively, we can plug in any number x and it would work. So then we would have infinite amount of solutions as well.

Triangle X(1, 6), Y(5, -2), Z(-5, -1) is mapped onto Δ

Δ

XʹYʹZʹ by a dilation with center (1, -2) and a scale factor of 3. Which function represents this dilation?

(x, y) → (1 + 3(x + 1), -2 + 3(y + 2))

(x, y) → (1 + 3(x - 1), -2 + 3(y - 2))

(x, y) → (1 + 3(x - 1), -2 + 3(y + 2))

(x, y) → (x + 3, y - 6)

30 points

Answers

On solving the provided question we can say that by herons formula, area of the triangle is, A = 2.828

What is triangle?

A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.

the provided points are Triangle X(1, 6), Y(5, -2), Z(-5, -1)

by herons formula

A = $\sqrt{s(s-a)(s-b)(s-c)}$

s = a+b+c/3

s = 5+2+5+/3 =12/3 = 4

Area of the triangle is, A =

$\sqrt{4(5-4)(4-2)(5-4)} \\A = \sqrt{4*1*2*1} \\A = \sqrt{8} \\A = 2\sqrt{2}\\ A = 2*1.414\\A = 2.828$

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Answer:

2.828

Step-by-step explanation:

Find the midpoint of the line segment with the given endpoints.
(-1, 5), (6,6)
A) (-3.5, -0.5)
C) (13,7)
B) (2.5, 5.5)
D) (2,6)

Answers

The answer should be (2,5) (11,2)

don't click on any links in this app example:I u⁣ploaded t⁣he a⁣nswer t⁣o a f⁣ile h⁣osting. H⁣ere's l⁣ink:

Answers

Answer:

I know someone answer my questions please quick!

Step-by-step explanation:

What is the value of c?
a)4 units
b)5 units
c)6 units
d)7 units

Answers

The value of c in the triangle is (b) 5 units

Finding the value of c in the triangle

From the question, we have the following parameters that can be used in our computation:

The right triangle

The length c is the hypotenuse of one of the triangles and can be calculated using the following Pythagoras theorem

c² = sum of squares of the legs

Using the above as a guide, we have the following:

c² = 3² + 4²

Evaluate

c² = 25

Take the square roots

c = 5

Hence, the hypotenuse of the right triangle is 5