A rectangle has a length of the cube root of 81 inches and a width of 3 to the 2 over 3 power inches. Find the area of the rectangle.

3 to the 2 over 3 power inches squared
3 to the 8 over 3 power inches squared
9 inches squared
9 to the 2 over 3 power inches squared

Answers

Answer 1
[tex]A= \sqrt[3]{81}*3^{ \frac{2}{3} }= \sqrt[3]{3^4}*3^{ \frac{2}{3} } = 3^{ \frac{4}{3} }*3^{ \frac{2}{3} }=3^{ \frac{4}{3} + \frac{2}{3} }=3^2=9 \ [/tex]

9 inches squared
Answer 2

Answer:

9 square inches.

Step-by-step explanation:

We have been given that a rectangle has a length of the [tex]\sqrt[3]{81}[/tex] inches and a width of [tex]3^{\frac{2}{3}}[/tex] power inches. We are asked to find the area of given rectangle.

We know that area of rectangle in length times width of rectangle.

[tex]\text{Area of rectangle}=\sqrt[3]{81}\times 3^{\frac{2}{3}}[/tex]

We can write 81 as [tex]3^4[/tex] as:

[tex]\text{Area of rectangle}=\sqrt[3]{3^4}\times 3^{\frac{2}{3}}[/tex]

Using exponent rule [tex]\sqrt[n]{a^m}=a^{\frac{m}{n}}[/tex], we can write [tex]\sqrt[3]{3^4}=3^{\frac{4}{3}}[/tex].

[tex]\text{Area of rectangle}=3^{\frac{4}{3}}\times 3^{\frac{2}{3}}[/tex]

Using exponent rule [tex]a^b\cdot a^c=a^{b+c}[/tex], we will get:

[tex]\text{Area of rectangle}=3^{\frac{4}{3}+\frac{2}{3}}[/tex]

[tex]\text{Area of rectangle}=3^{\frac{4+2}{3}}[/tex]

[tex]\text{Area of rectangle}=3^{\frac{6}{3}}[/tex]

[tex]\text{Area of rectangle}=3^{2}[/tex]

[tex]\text{Area of rectangle}=9[/tex]

Therefore, the area of given rectangle is 9 square inches.


Related Questions

which other angle must also measure 130°

Answers

opposite angles are identical so if angle 1 = 130

 than angle 3 is also 130 degrees

Answer:

Angle 3

Step-by-step explanation:

we know that

[tex]m<1=m<3[/tex] -----> by vertical angles

we have

[tex]m<1=130\°[/tex]

therefore

[tex]m<3=130\°[/tex]

What is the arc length of a circle that has a 6-inch radius and a central angle that is 65 degrees? Use 3. 14 for pi and round your answer to the nearest hundredth

Answers

C=2πr

C=2π6

C=37.6991118431

That's the arc length of the whole circle, i.e. the arc length of 360°.

65° is 18.055555555555555555555555555556% of 360°, (65/360*100)
so 18.055555555555555555555555555556% of 37.6991118431 is 
6.8067840827819444444444444444444.  Rounded to the nearest hundredth, the answer is 6.81 inches.

That's real pi, lets see if it makes a difference to use "stupid pi".

C=2πr

C=2π6

C=37.68

That's the arc length of the whole circle, i.e. the arc length of 360°.

65° is 18.055555555555555555555555555556% of 360°, (65/360*100)
so 18.055555555555555555555555555556% of 37.68 is 
6.803333333333333333333333.  Rounded to the nearest hundredth, the answer is 6.80 inches.
Yep makes a difference.  That's why you don't use stupid pi.  3.14159 is what we always used in engineering, or just the pi button and using a ton of digits.

Answer: 6.80 inches.

What is the equation of the line that is parallel to y=-2/3x+4 and that passes through (–2,–2)?

Answers
-
-
-
y=-2/3x-4/3
y=-2/3x-10/3
y=-2/3x-2/3
y=-2/3x-17/4

Answers

For lines to be parallel to one another they must have the same slope.  Our reference line has a slope of -2/3, so our parallel line is:

y=-2x/3+b, using the point (-2,-2) we can solve for "b", the y-intercept.

-2=-2(-2)/3+b

-2=4/3+b

-6/3-4/3=b

-10/3=b so our line is:

y=-2x/3-10/3  (or more neatly in my opinion :P)

y=(-2x-10)/3

It's the second one down from the top :)

Final answer:

The equation of the line parallel to y=-2/3x+4 and passing through (-2, -2) is y = -2/3x - 10/3.

Explanation:

To find the equation of a line parallel to y = -2/3x + 4 and passing through the point (-2, -2), we need to use the fact that parallel lines have the same slope. The given equation has a slope of -2/3, so the parallel line will also have a slope of -2/3. Using the point-slope form of a line, we can write the equation as:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope. Plugging in the values, we have:

y - (-2) = -2/3(x - (-2))

Simplifying the equation, we get:

y - (-2) = -2/3(x + 2)

y + 2 = -2/3x - 4/3

y = -2/3x - 4/3 - 2

y = -2/3x - 4/3 - 6/3

y = -2/3x - 10/3

Learn more about equation of a line here:

https://brainly.com/question/33578579

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The equation of a line is 2(y+1)=10x+3
The y-intercept of the line is ___, and the slope of the line is ___.

Answers

2y+2=10x+3
2y=10x+1
y=5x+1/2
y intercept is (0,1/2)
slope is 5

Answer:  The answer is 0.5 and 5.

Step-by-step explanation: The given equation of the line is

[tex]2(y+1)=10x+3.[/tex]

We are to find the y-intercept and the slope of the given line.

We know that the slope-intercept form of a line is given by

y = mx + c, where, 'm' is the slope and 'c' is the y-intercept of the line.

We have

[tex]2(y+1)=10x+3\\\\\Rightarrow 2y+2=10x+3\\\\\Rightarrow 2y=10x+3-2\\\\\Rightarrow 2y=10x+1\\\\\Rightarrow y=5x+0.5.[/tex]

Therefore, c = 0.5 and m = 5.

Thus, the y-intercept of the line is 0.5 and the slope is 5.

The expression 4 square root of 81^3 can be rewritten as_____.

A. 81^3/4

B.81^4/3

C. 81^12

D. 81^1/12

Answers

Answer:

81^1/12

HAVE A GREAT DAY

The expression ''4 square root of 81³'' can be rewritten as,

⇒ [tex]81^{\frac{3}{4} }[/tex]

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression to write is,

⇒ 4 square root of 81³

Now, It can be written as;

⇒ 4 square root of 81³

⇒ [tex]\sqrt[4]{81^{3} }[/tex]

By rule of exponent we get;

⇒  [tex]81^{\frac{3}{4} }[/tex]

Thus, The expression ''4 square root of 81³'' can be rewritten as,

⇒ [tex]81^{\frac{3}{4} }[/tex]

Learn more about the mathematical expression visit:

brainly.com/question/1859113

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If f(x) is an odd function, which statement about the graph of f(x) must be true?
It has rotational symmetry about the origin.
It has line symmetry about the line y = –x.
It has line symmetry about the y-axis.
It has line symmetry about the x-axis.

Answers

An odd function, by definition, is a function that is symmetric about the origin.

An even function, by definition, is a function that is symmetric with respect to the y-axis.

Since the question says that f(x) is an odd function, it has rotational symmetry about the origin. First option is correct.


ANSWER: symmetric about the origin.

Answer:It has rotational symmetry about the origin.

Step-by-step explanation:

An odd function : is a function that is symmetric about the origin.

An even function : is a function that is symmetric with respect to the y-axis.

Since , f(x) is an odd function, it has rotational symmetry about the origin.

its meaning that its graph remains unchanged after rotation of 180 degrees about the origin.

Therefore, It has rotational symmetry about the origin.

Linda is putting money into a savings account. She starts with $450 in the savings account, and each week she adds $70 .

Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Linda has been adding money. Write an equation relating S to W . Then use this equation to find the total amount of money in the savings account after 19 weeks.

Answers

450+added

S=450+70W would be the equation

then afet 19 weeks
W=19
S=450+70(19)
S=450+1330
S=1780



equation is
S=70W+450
after 19 weeks, has 1780
The equation is
S (w)=450+70w

The total amount of money in the savings account after 19 weeks is
S (19)=450+70×19
S (19)=1,780

Law of sines:

Triangle ABC has measures a = 2, b = 2, and m∠A = 30°. What is the measure of angle B?

15°
30°
45°
60°

Answers

It's B=30 degrees

since a and b are equal to number 2, they are the same value. Because A is 30 degrees, B is 30 degrees too.

Answer:  Second option is correct.

Step-by-step explanation:

Since we have given that

ΔABC has measures a=2, b=2, m∠A=30⁰

As we know the "Law of sines " i.e.

[tex]\frac{a}{\sin A}=\frac{b}{\sin B}\\[/tex]

so, we put the given values in above formula:

[tex]\frac{2}{\sin 30\textdegree}=\frac{2}{\sin B}\\\\\implies \sin 30\textdegree=\sin B\\\\\implies B=30\textdegreee[/tex]

Hence, Second option is correct.


The length and width of a rectangle are 4.9^9 cm and 5.3^3 cm, respectively. What is the approximate area of the rectangle, using only positive exponents?
A) 5^6cm^2
B) 4^6cm^2
C) 5^12cm^2
D) 4^12cm^2

Answers

approximate 
L = 4.9^9 cm = 5^9 cm
W = 5.3^3 cm = 5^3 cm

Area = L x W
Area = 5^9 x 5^3 = 5^12 cm^2

answer
C) 5^12 cm^2

Expand (2x-3y)^4 using Pascal's Triangle. Show work

Answers

(2x+3y)⁴
1) let 2x = a   and 3y = b

(a+b)⁴ = a⁴ + a³b + a²b² + ab³ + b⁴
Now let's find the coefficient of each factor using Pascal Triangle
     
                     0     |               1
                     1     |            1    1
                     2     |          1   2   1
                     3     |         1  3   3   1
                     4     |       1  4   6    4  1

0,1,2,3,4,.. represent the exponents of binomials 
Since our binomial has a 4th exponents, the coefficients are respectively:

(1)a⁴ + (4)a³b + (6)a²b² + (4)ab³ + (1)b⁴
Now replace a and b by their real values in (1):

2⁴x⁴ +(4)8x³(3y) + (6)(2²x²)(3²y²) + (4)(2x)(3³y³) + (1)(3⁴)(y⁴)

16x⁴ + 96x³y + 216x²y² + 216xy³ + 81y⁴

Answer:

16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4

Step-by-step explanation:

(2x - 3y)^4

Fifth line on a Pascal Triangle

1, 4, 6 4, 1

(1) 2x^4

2^4 = 16

2x^4 = 16x^4

16x^4

(4) 2x^3 (-3y)^1

2^3 = 8

-3^1 = -3

8 times -3 times 4 = -96

-96x^3y

(6) 2x^2 (-3y)^2

2^2 = 4

-3^2 = 9

4 times 9 times 6 = 216

216x^2y^2

(4) 2x^1 (-3y)^3

2^1 = 2

-3^3 = -27

2 times - 27 times 4 = -216

-216xy^3

(1) (-3y)^4

-3^4 = 81

81y^4

16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4

One bundle contains 500 $20 bills. what would be the total value of 44 bundles

Answers

We are given data that one bundle contains 500 of $20 bills. We are asked to determine the value (total) of 44 bundles. We can establish a formula for this:

let y = total amount
     x = number of bundles

y = $20 * 500x
y = $10,000 per bundle * x 

So, using this equation, we can determine the total amount of money by simply substituting the number of bundles that we have. 

To solve our problem, let x = 44 bundles

y = $10,000 per bundle * 44 bundles

y = $440, 000 

Therefore, the total amount is $440,000. 

Carol spends 17 hours in a 2-week period practicing her culinary skills. How many hours does she practice in 5 weeks?

Answers

Final answer:

Carol practices for 8.5 hours per week, so in a 5-week period, she would practice for a total of 42.5 hours.

Explanation:

The student asked how many hours Carol practices her culinary skills in a 5-week period, if she practices for 17 hours in a 2-week period. To find the answer, we calculate how many hours Carol practices per week by dividing the total hours she practices in two weeks by two. Then we multiply the weekly hours by the number of weeks in question, which is five.

The calculation is as follows: Carol practices for 17 hours / 2 weeks = 8.5 hours per week. Then, 8.5 hours/week x 5 weeks = 42.5 hours in total for a 5-week period.

Temperature dropped from 11 below zero to 4 below zero how much did the temperature drop

Answers

it would actually rise.

-11 + 4 = 6

-11 -10 -9 -8 -7 -6 -4 -3 -2 -1 0
   l ------------------->  l

it rose 6 degrees (or dropped -6 degrees, to match the question) :)

Find the equation of the quadratic function with zeros 10 and 14 and vertex at (12, -8).

Answers

we'll do the same we did on the previous one, using the vertex form.

our vertex is at (12, -8) thus h = 12, k = -8

and our zeros are 10, 0 and 14,0, so.. .we'll use say  hmm (10,0) to get the "a" coefficient.

[tex]\bf \qquad \textit{parabola vertex form}\\\\ \begin{array}{llll} \boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\ x=a(y-{{ k}})^2+{{ h}} \end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\ -------------------------------\\\\ vertex\ (12,-8)\ \begin{cases} h=12\\ k=-8 \end{cases}\implies y=a(x-12)^2-8 \\\\\\ \textit{we also know that } \begin{cases} y=0\\ x=10 \end{cases}\implies 0=a(10-12)^2-8 \\\\\\ 8=a(-2)^2\implies 8=4a\implies \boxed{2=a} \\\\\\ thus\qquad \boxed{y=2(x-12)^2-8}[/tex]

Solve by factoring and list only the positive solution: 2x2 - 5x = 88

Answers

It has to be noted that there are several ways to factor the item. One of these is shown below. 

The general form of a quadratic equation is,
            Ax² + Bx + C = 0

where A and B are numerical coefficients and C is the constant. If we are to express the given equation in this form, 
        2x² - 5x - 88 = 0

The sum of the roots, x₁ and x₂ is -B/A and the product is equal to C/A.

Sum: x₁ + x₂ = -(-5/2) = 5/2
 Product: x₁x₂ = -88/2 = -44

The values of x₁ and x₂ are 8 and -11/2.

The factors are (x - 8) and (x + 11/2)

ANSWERS: 8 and -11/2. 

Is it possible for a line segment to have more than one bisector?

Answers

Yes, it is possible to have more than one bisector in a line segment.

Bisector is a line that divides a line or an angle in to two equivalent parts. There are two types of Bisectors based on what geometrical shape it bisects.

Bisector of a Line Angle Bisector

 In general 'to bisect' something means to cut it into two equal parts. The bisector  is the one that doing the cutting process.

 

With a line bisector, we cut a line segment into two equal parts with another line - the bisector. Just imagine the line PQ is being cut into two equal lengths (PF and FQ) by the bisector line AB.

 

Whenever AB intersects at a right angle, it is called the "perpendicular bisector" of PQ. If it crosses at any other angle it is simply called a bisector. Drag the points A or B and see both types.

 

For obvious reasons, the point F is called the midpoint of the line PQ,

I SERIOUSLY NEED HELP HERE!!!!!


PLEASE SOMEONE HELP ME ON THIS!!!!!

NEED MAJOR HELP HERE CALCULATOR QUIT ON ME!!!!!!!
scientific calculator of a TI83 or TI84  ( does that help?)
Use the data below to find the correlation coefficient. (Remember to choose DiagnosticOn on your calculator.)




x        y
270   70
230   75
250   68
310   82
285   80
275   76
281   73
267   81
252   72
246   79

The correlation coefficient is _____. Round to the nearest thousandth.

THESE ARE MY OPTIONS:
a. 0.438
b. 0.192
c. 0.5
d. 0.720

Answers

The plot of the data and the regression line is shown in the figure below.
The best straight line is
y = 51.0686 + x 0.092x

The correlation coefficient is 0.192.

Answer: b

Multiply 3 [ 1 5 -5 6 0 0 ]

Answers

Simply multiply the number outside the brackets with each one inside it..

3 [ 1 5 -5 6 0 0 ]
3 x 1 = 3
3 x 5 = 15
3 x -5 = -15
3 x 6 = 18
3 x 0 = 0
3 x 0 = 0
[ 3 15 -15 18 0 0 ]

[ 3 15 ]
[ -15 18 ]
[ 0 0]

The answer is B and I hope I explained this well for you.

Find the equation for the tangent line of f(x)=−3x2−7x+3 at x=3.

Answers

hello : 
the equation for the tangent line at point : A(x' , y' ) is :
y - y' = f'(x') (x-x')........f'(x') the slope
in this exercice : 
f(x) = -3x²-7x+3  and : x' = 3
y' = f(3) = -3(3)²-7(3) +3 =- 45
but : f'(x) = -6x-7   ...( derivate of : f(x) )
f'(3) = -6(3) -7 = - 25

the equation for the tangent line is : y - (-45) = -25(x-3)
y = -25x+30

2x-5y=-6; 2x-7y=-14

Answers

2x-5y = -6
2x-7y = -14

-2x+5y = 6
2x - 7y = -14

-2y= - 8
y = 4

What equation is solved by the graphed systems of equations? Two linear equations that intersect at the point negative 1, negative 4.

Answers

To solve this problem, we have to manually solve for the value of x for each choices or equations. The correct equation will give a value of -1 since the linear equations intersects at point (-1, -4).

1st: 7x + 3 = x + 3

7x – x = 3 – 3

6x = 0

x = 0                (FALSE)

 

2nd: 7x − 3 = x – 3

7x – x = 3 – 3

6x = 0

x = 0                (FALSE)

           

3rd: 7x + 3 = x − 3

7x – x = - 3 – 3

6x = -6

x = -1               (TRUE)

 

4th: 7x − 3 = x + 3

7x – x = 3 + 3

6x = 6

x = 1                (FALSE)

 

Therefore the answer is:

7x + 3 = x − 3

In this exercise, we are going to solve using our knowledge of systems and in this way we will find that the equation that satisfies the points.

As we know that the equation that will satisfy will have to have the values ​​of X=-1, we will solve each one of the alternatives as:

First equation is:

[tex]7x + 3 = x + 3\\6x = 0\\x = 0[/tex]

We realize that the value of x is not what we want so it doesn't satisfy us.

second equation is:

[tex]7x - 3 = x - 3\\7x- x = 3- 3\\6x = 0\\x = 0[/tex]

We realize that the value of x is not what we want so it doesn't satisfy us.

         

third equation is:

[tex]7x + 3 = x − 3\\6x = -6\\x = -1[/tex]

fourth equation is:

[tex]7x − 3 = x + 3\\7x – x = 3 + 3\\6x = 6\\x = 1[/tex]

We realize that the value of x is not what we want so it doesn't satisfy us.

See more about systems at brainly.com/question/7589753

place a square on a coordinate graph and label each vertex with variables. prove that the diagonals of a square are congruent and perpendicular to each other.

Answers

Final answer:

To prove that the diagonals of a square are congruent and perpendicular, label the vertices of a square on a coordinate grid and calculate the slopes and lengths using the slope formula and distance formula respectively. The diagonals have slopes of +1 and -1, proving they are perpendicular, and they have equal lengths, proving they are congruent.

Explanation:

To prove that the diagonals of a square are congruent and perpendicular, we place a square with its vertices on a coordinate grid and label each vertex with variables.

Let's consider a unit square where c = 1 for simplicity, which means the length of each side is 1 unit. Place the square so that one vertex is at the origin (0,0), and label the vertices A(0,0), B(1,0), C(1,1), and D(0,1).

The diagonal AC will have endpoints at A(0,0) and C(1,1), and diagonal BD will have endpoints at B(1,0) and D(0,1). The slope of diagonal AC is (1 - 0)/(1 - 0) = 1, and the slope of diagonal BD is (1 - 0)/(0 - 1) = -1. Since the product of their slopes is -1 (1 * -1 = -1), this proves that they are perpendicular to each other.

To show they are congruent, we calculate their lengths using the distance formula: the distance between two points (x1,y1) and (x2,y2) is √[(x2 - x1)² + (y2 - y1)²]. Applying this to AC and BD reveals both lengths to be √[(1-0)² + (1-0)²] = √[1 + 1] = √2, proving the diagonals are congruent.

What is a rule for the total cost of the tickets ? Give the rule in words and as a algebraic expression

Answers

from the looks of it, each ticket costs $ 12...and there is a fixed rate of 150

ur equation would be : y = 12x + 150....where x is the number of students and y is the total cost...but that is an equation, and u want an expression...so I guess the expression would be 12x + 150

the cost of each ticket (12) multiplied by the number of students (x), added to 150 will give u the total cost


ok, it seems that it is multiplying the number of studnets by 12 then adding 150

so theh rule is 12s+150 where s is the number of students

What number must be added to the expression below to complete the square?

x^2+3x

A. 9
B. 9/4
C. 3/2
D. 3

Answers

B. (b/2)^2 is 9/4. There

I think it is c or D but it should be C


hope that help

[I don't think it did lol ]


BMK

Alex has been serving 2/3 cup of lemonade to each student. If he has 1 1/3 cups of lemonade left, how many students can still get lemonade?

Question 2 options:

1


2


3


0

Answers

Answer: 2
In this case, every student is served 2/3 cup of lemonade and Alex has 1 1/3 cups of lemonade left. Then you just need to divide the lemonade left with lemonade served per student. The calculation would be:

Number of students served= lemonade left / lemonade served per student
Number of students served= 1 1/3 cup / (2/3 student/cup) = 2 student

Daria applied a transformation to triangle ABC to obtain triangle A′B′C′. The two triangles are not congruent. Which of the following could be the transformation Daria applied?

Answers

Daria could have applied dilation.

What is the area of the composite figure?

(6π + 4) cm2
(6π + 16) cm2
(12π + 4) cm2
(12π + 16) cm2

Answers

I hope this helps you



in this figure there are 3 semi circle and one square, one semi circle radius is 2 , square all sides 2.2=4

one semi circles area =pi.r²/2


one semi circle area = pi.2²/2=2.pi


three semi circle area =3.2.pi =6.pi


square area = 4² =16


area of composite figure = 6.pi+16

The area of the composite figure is 6π + 16 cm²

Composite Figure:

Composite figures are composed of different dimensional figures. The area of a composite figure is the sum of the whole 2 dimensional figures that forms the composite figure.

Therefore, the figure above has 3 semi circle and 1 square.

Therefore, the area can be calculated as follows;

area = sum of the area of the 3 semi circle + area of the square

area = 1 / 2 πr² + 1 / 2 πr² + 1 / 2 πr² + L²

area = 3 / 2 (πr²) + L²

where

r = 2 cm

L = 4 cm

Therefore,

area of the composite figure = 3 / 2(π × 4) + 4²

area of the composite figure = 3 / 2(4π) + 16

area of the composite figure =  6π + 16 cm²

learn more on composite figures here: https://brainly.com/question/1639299

What is the area of the region completely bounded by the curve y=-x^2+x+6?

Answers

Firstly, factorise the equation:
y = -1 (x² - x - 6)
y = -1 (x - 3)(x + 2)
From this, we can tell the x-axis intercepts are -2 and 3.
(To do this, simply equate any of the expressions involving x to 0 and rearrange to give x).

Now, integrate between the limits -2 and 3:
R = area of region bounded
R = ³∫₋₂ -x² + x + 6 .dx
R = ³[-1/3x³ + 1/2x² + 6x]₋₂
R = (-1/3(3)³ + 1/2(3)² + 6(3)) - (-1/3(-2)³ + 1/2(-2)² +6(-2))
R = (-9 + 9/2 + 18) - (8/3 + 2 - 12)
R = (27/2) - (-22/3)
R = 27/2 + 22/3 = 125/6 units²
Final answer:

To find the area of the region completely bounded by the curve y=-x^2+x+6, you can integrate the equation with respect to x and evaluate it between the x-values where the curve intersects the x-axis. By solving the quadratic equation -x^2+x+6=0, you can determine the x-values. Then, evaluate the definite integral between these x-values to find the area.

Explanation:

The area of the region completely bounded by the curve y=-x^2+x+6 can be found by integrating the equation with respect to x and evaluating it between the appropriate bounds. The integral of the given equation is ∫(-x^2+x+6) dx. To find the area, we need to find the definite integral between the x-values where the curve intersects the x-axis. First, set the equation equal to zero and solve for x:

-x^2+x+6=0

This quadratic equation can be factored as: (x-2)(x+3). Therefore, the curve intersects the x-axis at x=2 and x=-3.

By evaluating the definite integral between x=-3 and x=2, we can find the area of the region:

Area = ∫-32 (-x^2+x+6) dx

Integrating this equation will give you the area of the region bounded by the curve y=-x^2+x+6.

Help? @texaschic101
Parabola and its vertex

Answers

Formula is y = a(x-h)^2 + k

f (x) = a(x-4)^2 - 1

15 = a(0-4)^2 - 1

15 = a(-4)^2 - 1

15 = a(16) - 1

15 +1 = 16a

16 = 16a

16/16 = 16a/16

1 = a

A must be equal to 1

y = (x-4)^2 - 1

y = x^2 - 8x + 16 - 1

y = x^2 - 8x + 15

Factor (find two number that summed give -8 and multiplied +15)

(x-3)(x-5) = 0

(x-3) = 0 -> x = 3

(x-5) = 0 -> x = 5

So answer is: (3;0) and (5;0)

Assume a plane is flying directly north at 200 mph, but there is a wind blowing west at 23 mph. Part I: Express both the velocity of the plane and the velocity of the wind as vectors, using proper notation to represent each direction of motion. Part II: What is the velocity vector of the plane? Part III: What is the ground speed of the plane?

Answers

Define i  as a unit vector in the eastern direction.
Define j as a unit vector in the northern direction.

Part I
Because the wind is blowing west, its velocity vector is
 -23i mph or as (-23, 0) mph
Because the plane is traveling north, its velocity vector is
  200j mph or as (0, 200) mph 

Part II
The actual velocity of the plane is the vector sum of the plane and wind velocities.
That is,
200j - 23i or (-23, 200) mph

Part III
The ground speed of the plane is the magnitude of its vector.
The ground speed is
√[200² + (-23)²] = 201.32 mph

The ground speed of the plane is 201.3 mph (nearest tenth)

Not:
The direction of the plane is
 tan⁻¹ 23/200 = 6.56° west of north.


Final answer:

The velocity of the plane is 200 mph due north and the velocity of the wind is 23 mph due west. The velocity vector of the plane is 200 mph due north minus 23 mph due west. The ground speed of the plane can be found using the Pythagorean theorem.

Explanation:

Part I: The velocity of the plane can be represented as 200 mph due north, and the velocity of the wind can be represented as 23 mph due west.

Part II: To find the velocity vector of the plane, we subtract the velocity of the wind from the velocity of the plane. The resultant velocity vector of the plane is 200 mph due north minus 23 mph due west.

Part III: The ground speed of the plane is the magnitude of the resultant velocity vector of the plane. We can calculate it using the Pythagorean theorem: ground speed = square root of (200^2 + 23^2).

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