A rectangle has one side on the xx-axis and two vertices on the curve y=74+x2. y=74+x2. find the vertices of the rectangle with maximum area.

Answers

Answer 1

Given that a rectangle has one side on the x-axis and two vertices on the curve [tex]y= \frac{7}{4+x^2} [/tex]

We find the vertices of the rectangle with maximum area as follows:

Because of the symmetry of the given curve, the base of the retangle is 2x and the height of the rectangle is y.

Thus, the area of the required rectangle is 2xy where [tex]y= \frac{7}{4+x^2} [/tex]

Thus, [tex]Area=\frac{14x}{4+x^2}[/tex]

For maximum area, we find value of x for which the derivative of the function for the area is equal to zero.

The derivative of the function for the area is given by:

[tex](Area)'= \frac{14(4-x^2)}{(4+x^2)^2} =0 \\ \\ \Rightarrow4-x^2=0 \\ \\ \Rightarrow x^2=4 \\ \\ \Rightarrow x=\sqrt{4}=2[/tex]

Thus, the area is maximum when x = -2 and x = 2

When x = -2,

[tex]y= \frac{7}{4+(-2)^2}= \frac{7}{4+4} = \frac{7}{8} [/tex]

Similarly, when x = 2,

[tex]y= \frac{7}{4+(2)^2}= \frac{7}{4+4} = \frac{7}{8} [/tex]

Therefore, the required vertices are: [tex](-2,0),\ \left(-2, \frac{7}{8} \right), \ \left(2, \frac{7}{8}\right), \ (2, 0)[/tex]

Answer 2

Final answer:

The rectangle with maximum area does not exist as the curve y = 74 + x^2 does not intersect the x-axis.

Explanation:

To find the vertices of the rectangle with maximum area, we need to find the points of intersection between the curve y = 74 + x^2 and the x-axis.

Setting y = 0, we can solve the equation 74 + x^2 = 0 to find the x-coordinates of the vertices.

Since the curve is a parabola facing upwards, there are no points of intersection with the x-axis.

Therefore, the rectangle does not exist.

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Related Questions

Please help thank you.

30 points.

Answers

Question 1:

The sum of all angles with a polygon with n sides is
180(n-2)

We have 20 sides.

180(20 -2) = 180 * 18
= 3240

Thus, the sum of all angles is 3240 degrees.
The second choice is your answer.

Question 2:

First, let's calculate the sum of all the angles.

180(12 - 2) = 1800

Now, divide that by the number of sides to get the measure of one angle.

1800/12 = 150

Your answer is the last choice.

Question 3:

Do the same as we did in question 2, but instead of 12 sides, there are 6 sides.
A regular hexagon has an interior angle measure of 120 degrees.
(Note: you should remember that for future reference).

Now, let's form the following equation

3x = 120

Divide both sides by 3

x = 40

The first choice is your answer.

Have an awesome day! :)

Hey there!

1. 3240. 20 gon is called an isogon
2.150 because the formula for the interior angle or regular polynomials is [tex] \frac{180(n-2)}{n}[/tex]
3.3x = 120
Divide both sides by 3
x = 40

The sum of two integers is 28, and their product is 192. what are the two integers?

Answers

The two integers are 12 and 16.

The product of 12 and 16 is 192, and the sum is 28.

State a true conclusion. 1) If a ● b = 0, then a = 0 or b = 0 2) 3x = 0

Answers

Proposition 1) If a ● b = 0, then a = 0 or b = 0

Proposition 2) 3x = 0

=> 3 = 0 or x = 0

Given that 3 ≠ 0, x must be equal to 0.

Conclusion: x = 0

Answer:

x=0

Step-by-step explanation:

Proposition 1) If a ● b = 0, then a = 0 or b = 0

Proposition 2) 3x = 0

=> 3 = 0 or x = 0

Given that 3 ≠ 0, x must be equal to 0.

Conclusion: x = 0

Can someone how to get the answer? Thanks! PLEASE HELPpppppppppppppppppppppppp

Answers

[tex]\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right) \ \textgreater \ \ln \left(6^x\right)=\ln \left(21\right)[/tex]

[tex]\mathrm{Apply\:log\:rule}:\ \log _a\left(x^b\right)=b\cdot \log _a\left(x\right) \ \textgreater \ \ln \left(6^x\right)=x\ln \left(6\right) [/tex]

[tex]x\ln \left(6\right)=\ln \left(21\right) \ \textgreater \ \mathrm{Divide\:both\:sides\:by\:}\ln \left(6\right) \ \textgreater \ \frac{x\ln \left(6\right)}{\ln \left(6\right)}=\frac{\ln \left(21\right)}{\ln \left(6\right)} [/tex]

[tex]Therefore \: x=\frac{\ln \left(21\right)}{\ln \left(6\right)}[/tex]

In your case ln means [tex] \frac{log 21}{log 6} [/tex]

in your graphing calculator you have to press the button (2nd) then (catalog) which is the number (0).
after that you press (ALPHA) (L) which iOS the button for the parenthesis (. ). )
find LOG and then plug in the number you want.
repeat this until you get the equation you want.

I have a box of nickels and dimes. When I counted the money, I was surprised to see that I have exactly $20. Then it occurred to me that, if my nickels were dimes and my dimes were quarters, I would have $46.25. How many dimes and nickels are in the box?

Answers

It depends. There could be ten dimes and twenty nickels or one dime and thirty-nine nickels. Maybe word this differently.

There are 150 nickels and 125 dimes in the box.

Let x = the number of nickels in the box

Let y = the number of dimes in the box

Firstly, we have to convert value of nickels and dimes into cents.

Value of 1 nickel = 5 cents

Value of 1 dime = 10 cents

Value of 1 dollar = 100 cents

Thus, the equation (1) (in cents) can be written as:

x + 2y = 400 Equation (1)

If my nickels were dimes and dimes were quarters:

A quarter is worth 25 cents.

Thus, the equation (2) (in cents) can be written as:

5x+10y =2000

5(x+2y)= 2000

x+2y = 2000/5

2x + 5y = 925 Equation (2)

Multiply Equation (1) by "2", we get:

10x +25y= 4625

5(2x+5y)=4625

2x+5y=4625/5

2x + 4y = 800 Equation (1)

Subtraction Equation (1) from Equation (2):

2x +5y-2x -4y=925-800

y= 125

2x + 5y = 925

2x + 5 × 125 = 925

2x + 625 = 925

2x = 925 - 625

2x = 300

x = 150

There are 150 nickels and 125 dimes in the box.

Here, we can check it by putting the value of x and y in Equation (1):

5x + 10y = 2000

5× 150 + 10 × 125 = 2000

750 + 1250 = 2000

2000 = 2000

There are 150 nickels and 125 dimes in the box.

What is the probability of throwing a number greater than 4 when rolling a number cube that has sides labeled 1, 2, 3, 4, 5, 6?

Answers

What is probability rolling a number greater than 4?

The only numbers there are is 5 and 6.
That means that there are 2 outcomes out of 6 total outcomes.
That would be 2/6.
Divide the the top and bottom by 2.
In simplest form it would be 1/3.
2/6=1/3

The answer is 1/3. The probability of throwing a number greater than 4 is 1/3.

Final answer:

The probability of rolling a number greater than 4 on a six-sided die is 1/3 or 0.33 or 33%. This is obtained by dividing the number of favourable outcomes (2, i.e., 5 and 6) by the total number of outcomes (6).

Explanation:

The subject of the question is the probability in mathematics, specifically related to an experiment involving rolling a six-sided die. To answer your question, a six-sided die has numbers from 1 to 6. But you are asking about throwing a number greater than 4. There are only two outcomes that are greater than 4, the numbers 5 and 6. Therefore, the possible favorable outcomes are 2.

The total outcomes (since it's a six-sided standard die) are 6. Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. In this case, it will be 2/6 = 1/3 which is approximately 0.33 or 33%.

This means, if you roll the die, on an average, you can expect to roll a number greater than 4 about 33 times if you roll it 100 times.

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For all real numbers a and b if a not equal to 0 and b not equal to 0 then a^2 + b^2 does not equal (a+b)^2

Answers

no the answer would be (a+b)^4 hope this helps you out

What is the value of x in the parallelogram?

Answers

The value of x is (180-67) = 113 degrees.

Extra information:

In a parallelogram, the consecutive angles are supplementary (so in this case, that means that because the line from corner x to the corner of 67 degrees is 180 degrees in total, the value of x has to be 180-67 = 113 degrees).

Which pair of triangles are congruent by ASA ?

Answers

the answer for this question is D because it has an angle then a side measurement then it has another angle measurement

The pair of triangles that are congruent by ASA is figure C

The ASA theorem

Two triangles can be judged to be congruent using the ASA (Angle-Side-Angle) criterion. For triangles to satisfy ASA congruency:

Angle: The triangles' two matching angles have to coincide.Side: There must also be congruence on the included side that lies between the congruent angles.

Put otherwise, the ASA criterion determines that two triangles are congruent if two of their angles and the included side are congruent, respectively. For this criterion, the angles and sides' order matters.

Based on the explanations, hence the pair of triangles that are congruent by ASA is figure C

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Let p: The shape is a rhombus. Let q: The diagonals are perpendicular. Let r: The sides are congruent. Which represents "The shape is a rhombus if and only if the diagonals are perpendicular and the sides are congruent”?

Answers

Given that p represents "The shape is a rhombus", q represents "The diagonals are perpendicular", and r represents "The sides are congruent".

"if and only if" is represented using the biconditional logic operator (↔) and "and" is represented using the logical conjuction operator (∧).

Therefore, "The shape is a rhombus if and only if the diagonals are perpendicular and the sides are congruent” is represented by "p ↔ (q ∧ r)"

Answer:

C.)p ↔ (q ∧ r)

Step-by-step explanation:

I scored a 100 on my quiz

For what values of x and y must each figure be a parallelogram?

Answers

(5x - 180)° +x° = 180°

(6x - 180)° = 180°

6x = 360°

x = 60°

5(60)-180= 300-180=120°

Use the graph shown to fill in the blank.

When x = -3, then y = a0.

Answers

All you need to do is read the value of y that corresponds to x=-3
you will find that:
y= -6 at x= -3

(0, -3)(-1,-4)

slope = (-3+4)/(0+1) = 1 and b = -3
so y = 7x - 3
when x= -3 then
y = 1(-3) -3
y = -3 - 3
y = -6

answer
When x = -3, then y = -6

Important state labor laws and safety regulations came about as a result of the 1911 __________.
a. labor relations act
b. triangle shirtwaist factory fire
c. homestead strike
d. massachusetts child safety campaign

Answers

b. triangle shirtwaist factory fire

Answer:

Option B is correct.

Step-by-step explanation:

Important state labor laws and safety regulations came about as a result of the 1911 - Triangle shirtwaist factory fire

The Triangle shirtwaist factory fire was one of the worst industrial disasters during that time. Many people died due to a fire breakout in the factory. They could not escape the building and died due to smoke inhalation or burns.

After this incident, a Committee on Public Safety was formed that led to a legislation requiring labor laws and improved factory safety standards.

A triangle has base 6 cm and area 42 cm^2 what is the height of the triangle

Answers

To find the area of a triangle, the formula is
A=bh/2
42=6h/2
Divide both sides by 6 giving you
h/2=7
Then, multiply by 2 on both sides giving you
h=14
The height then is 14 cm.

Final answer:

The height of the triangle is 14 cm.

Explanation:

To find the height of the triangle, we can use the formula for the area of a triangle: Area = 1/2 * base * height. In this case, the base is 6 cm and the area is 42 cm^2. By rearranging the formula, we can solve for the height: Height = 2 * Area / Base. Plugging in the values, we get Height = 2 * 42 cm^2 / 6 cm = 14 cm.

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Rob cuts 15-foot wire into 8 equal pieces. About how long is each piece

Answers

The technical answer is 1.875
but about would be 2 ig

Heyo!

The answer is 1.875 ft.

You can solve this by dividing 15 ft by 8 to get your length per piece.

Help please! I need to show my work

Answers

The centroid of a triangle is 2/3 of the way from the vertex to the opposite side. The centroid is the intersection of the medians. Each median has as endpoints a vertex and the midpoint of the opposite side.
a.
PC is 2/3 of PY, and CY is 1/3 of PY.
If CY = 10, then PC = 20, and PY = 30

b.
If QC = 10, then ZC = 5, a nd ZQ = 15.

c.
X is the midpoint of side PQ.
PX = XQ = 1/2(PQ)
If PX = 20, PQ = 40

The table below represents the atmospheric temperature at a location as a function of the altitude:

Altitude
(in thousand feet)
x
15
20
25
30

Temperature
(in °C)
f (x)

4
−6
−16
−26

The average rate of change of the function between x = 15 to x = 25 is ___degrees Celsius per thousand feet and represents the rate of change of temperature per thousand feet.

Answers

from x15 to x 25 is 4 - -16 = -20 degree change

25 -15 = 10000 feet

-20/10 = -2 degrees every 1000 feet

Answer: The average rate of change of the function between x = 15 to x = 25 is -2 degrees Celsius per thousand feet and represents the rate of change of temperature per thousand feet.

Step-by-step explanation:

The average rate of change of a function y=f(x) from [tex]x_1\ to\ x_2[/tex]is given by :-

[tex]k=\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{-6-4}{20-15}[/tex]

Now, by the given table the average rate of change of the function between x = 15 to x = 25 will be :_

[tex]k=\frac{-16-4}{25-15}=\frac{-20}{10}=-2[/tex]

Hence, the average rate of change of the function between x = 15 to x = 25 is -2 degrees Celsius per thousand feet and represents the rate of change of temperature per thousand feet.

What is the slope of a line that is perpendicular to a line whose equation is −2y=3x+7 ?

Answers

find the slope of the line given:

y = mx+b

y = -3x/2 -7/2

slope = -3/2

the slope of the perpendicular line is the negative reciprocal

so answer is 2/3

The answer is 0.66, or 2/3. Graph the line, and then use a protractor to create a line perpendicular to the line. Then, calculate the slope of the line.

Find the quotient of 3 2/5 and 3 2/6. Express your answer in simplest form.

Answers

32/5÷32/6=32/5×6/32=6/5

[02.05]
Solve for x: 4 − (x + 2) < −3(x + 4) (1 point)

x < −7

x > −7

x < −9

x > −9

Answers

first, get rid of the paranthesis:
4-x-2<-3x-12 => 2-x<-3x-12
next, eliminate x on one side. here I'll eliminate -3x by adding 3x on both side:
2+2x<-12
2x<-14
x<-7

Option A

x<-7

Given :

[tex]4 - (x + 2) < -3(x + 4)[/tex]

Explanation :

Solve the given inequality for x by applying algebraic properties

We use order of operation to simplify the left and right side of inequality

[tex]4- (x + 2) < -3(x + 4)\\4-x-2<-3x-12\\-x+2 < -3x -12\\Add\; 3x \; on \; both \; sides\\2x+2<-12\\Subtract\; 2\; from \; both \; sides\\2x<-14\\Divide \; both \; sides \; by 2\\x<-7[/tex]

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Which graph represents the solution set of the system of inequalities?

Answers

Answer:
(see image)
bottom right image

Explanation:
First try the origin (0,0) to rule out two of the graphs.
3y ≥ x - 9
3(0) ≥ (0) - 9
3 ≥ - 9 yes
3x + y > - 3
3(0) + (0) > - 3
3 > - 3 yes
so the origin should be in the shaded area of the graph, which rules out the top right and bottom left graphs.
Now try a coordinate that is in the shaded area of one of the remaining graphs, but not in the other one. If it works, the graph is the one that has that point in the shaded region, and vice versa.
Try point (4, 2)
3y ≥ x - 9
3(2) ≥ (4) - 9
6 ≥ - 5 yes
3x + y > - 3
3(4) + (2) > - 3
12 + 2 > - 3
14 > - 3 yes
So the graph is the bottom right one since (4, 2) is included in that shaded region.

What is the value of cos 45 degrees?
(answers in picture)

Answers

The answer will be b

What is the equation of the line shown in this graph?
pls help

Answers

the line of the graph is vertical since both x coordinates are the same ( both are -2)

so the equation would be x = -2

a line that passes through two sides of triangles divides the sides proportionally

A. always
B. sometimes
C. never

Answers

The answer is B: Sometimes

The possible answer is B: sometimes so good luck

which graph shows the inequality y>-2

Answers

The answer for your question is b

The expression 3x² represents

Answers

Final answer:

The expression 3x² signifies a quadratically powered variable (x) that is multiplied by the coefficient 3, commonly seen in algebraic equations.

Explanation:

The expression 3x² represents a mathematical formula where the variable x is squared (multiplied by itself) and then multiplied by 3. The squaring of a variable is a fundamental concept in algebra that symbolizes a number raised to the power of 2.

The coefficient 3 in this case indicates that the square of x is being tripled. Therefore, if x is 2, x² would be 4, and 3x² would be 12. This type of expression is essential in solving quadratic equations and appears frequently in various mathematical contexts.

Difference of two numbers is 8. find the smallest possible product.

Answers

The smallest possible product would be 8. The two numbers used are 0 and 8. If you subtract 0 from 8 you get the difference of 8. If you add 8 plus 0, you get the product of 8.

What theorem or postulate can prove a triangle to be an isosceles?

Answers

Final answer:

The Converse of the Base Angles Theorem states that if two angles of a triangle are congruent, then the triangle is isosceles.

Explanation:

In geometry, the theorem that can prove a triangle to be isosceles is the Converse of the Base Angles Theorem. It states that if two angles of a triangle are congruent, then the triangle is isosceles. In other words, if the two base angles of a triangle are equal, then the triangle is isosceles. For example, if angles A and B in triangle ABC are congruent (angle A = angle B), then triangle ABC is isosceles.

I have a graphing question

Answers

The point slope form of the equation of a straight line is given by

[tex](y-y_1)=m(x-x_1)[/tex]

where [tex](x_1,\ y_1)[/tex] is a point on the line.

Given

[tex]y-3=\frac{3}{4}(x+2) \\ \\ y-3=\frac{3}{4}(x-(-2))[/tex]

Thus the line has a slope of 3/4 and passes through the point (-2, 3), we can obtain another point on the line by substituting an arbitrary value for x.

Consider x = 2, then

[tex]y-3= \frac{3}{4} (2+2)= \frac{3}{4} (4)=3 \\ \\ \Rightarrow y=3+3=6[/tex]

Thus another point on the line is (2, 6).

The line can then be graphed by joining the two points.

The vertex of the graph of a quadratic function is (–2, –9) and the y-intercept is (0, –5). Which of the following is the equation of the function? A. y = x2 +10x + 2 B. y = x2 +10x – 2 C. y = x2 – 2x – 5 D. y = x2 + 4x – 5

Answers

using up extra characters

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