the discriminant of a quadratic equation is negative. one solution is 2+3i. what is the other solution?

Answers

Answer 1
negative means it has 2 non-real roots

we assume that the coefients of the equation are real so therefor

if a+bi is a root then a-bi is also a root

if 2+3i is a root then 2-3i is also a root (solution)


the other solution is 2-3i
Answer 2

Answer:

2-3i

Step-by-step explanation:


Related Questions

5 radians is the same as _____.

Answers

There are 2π radians per 360° so

5rad(360°/2π rad)=900/π degrees

That is approximately:

286.48° (to nearest hundredth of a degree) or

286°28'44" (to nearest second)


Answer:

286°28'44

Step-by-step explanation:

The function h(x) = x2 + 14x + 41 represents a parabola.
Part A: Rewrite the function in vertex form by completing the square. Show your work. (6 points)
Part B: Determine the vertex and indicate whether it is a maximum or a minimum on the graph. How do you know? (2 points)
Part C: Determine the axis of symmetry for h(x). (2 points)

Answers

Part A:

[tex]h(x)= x^{2} +14x+41[/tex]

The first step of completing the square is writing the expression [tex] x^{2} +14x[/tex] as [tex] (x+7)^{2} [/tex] which expands to [tex] x^{2} +14x+49[/tex].

We have the first two terms exactly the same with the function we start with: [tex] x^{2} [/tex] and [tex]14x[/tex] but we need to add/subtract from the last term, 49, to obtain 41. 

So the second step is to subtract -8 from the expression [tex] x^{2} +14x+49[/tex]

The function in completing the square form is
[tex]h(x)= (x+7)^{2}-8 [/tex]

Part B:

The vertex is obtained by equating the expression in the bracket from part A to zero

[tex]x+7=0[/tex]
[tex]x=-7[/tex]

It means the curve has a turning point at x = -7

This vertex is a minimum since the function will make a U-shape. 
A quadratic function [tex]a x^{2} +bx+c[/tex] can either make U-shape or ∩-shape depends on the value of the constant [tex]a[/tex] that goes with [tex] x^{2} [/tex]. When [tex]a[/tex] is (+), the curve is U-shape. When [tex]a[/tex] (-), the curve is ∩-shape

Part C:

The symmetry line of the curve will pass through the vertex, hence the symmetry line is [tex]x=-7[/tex]

This function is shown in the diagram below




to anyone who may be a lil' sus, the previous answer is correct.

SOMEONE PLEASE HELP ME WITH MATH! WILL GIVE BRAINLIEST TO FIRST PERSON WHO ANSWERS!!

Solve each system by substitution.
1. -5x-8y=9
y= -3.

2. y= -3x
-6x-2y=3

3. 6x+5y=10
y=-8x+2

4.6x+2y= -6
y= -3x -3

Thank you!

Answers

1. -5x-8(-3)=9
-5x+24=9
-5x=-15
x=3
(3,-3)

2. -6x-2(-3x)=3
-6x+6x=3
0=3
No solution

3. 6x+5(-8x+2)=10
6x-40x+10=10
-34x+10=10
-34x=0
x=0
y=-8(0)+2
y=2
(0,2)

4. 6x+2(-3x-3)=-6
6x-6x-6=-6
-6=-6
Infinitely many solutions

Angles θ and Φ are supplementary, and theta equals 3pi/7 find the measure of theta

Answers

hello : 
Φ + 3π/7 = π 
Φ =  π - 3π/7
Φ =4π/7 

From Plato: Φ =7π/7 or π.

Walter bought a square plot of land. the perimeter is 544 feet. how wide is the property?

Answers

perimeter of a square = is side times 4

so divide 544 by 4

544/4 = 136

the property is 136 feet wide

How can you eliminate the fraction from each side of the equation to make finding the
solution easier?
The equation is 1/2(x - 4) = 1/3x + 2

Answers

hello : 
 1/2(x - 4) = 1/3x + 2
multiply by : 6
3(x-4) = 2x +4
solution easier:
3x -12 = 2x +4
3x - 2x = 12+4
 x = 16

if 6a-8b=0 and c=12b, the ratio of a to c is equivalent to the ratio of one to what number?

Answers

6a=8b
3a=4b
c=3*(4b)=3*3a=9a
c=9a
First find the ratio of a:b since you know 6a=8b, then reduce to get 3a=4b. Since c=12b, let's multiply everything by 3 so 9a=12b. This means 9a=c. This is the same as 9a=1c so now you can easily get your answer:)

Juanita’s boss told her that once she has worked with the company for 10 years, her salary will be increased to 3 times what she makes now plus $2,000. If s represents her original salary, which expression represents what her salary will be after 10 years at the company?

Answers

The expression represents what her salary will be after 10 years at the company is
A=3s+2000

Answer:3s+2,000

Step-by-step explanation:

An amusement park is installing a new roller coaster. The park intends to charge $5 per adult and $4 per child for each ride. It hopes to earn back more than the $650,000 cost of construction in four years. With the best of weather, the park can provide 125,000 adult rides and 50,000 child rides in one season

Answers

Keeping the price and amount of people in mind, the answer is actually pretty easy to find. Just multiply 125,000 by 5, which equals 625,000 and 50,000 by 4, which equals 200,000, and add the products together to get $825,000 total income per season.

Answer:

Step-by-step explanation:

5x+ 4y greater than or equal to 650,000

500,000

y greater than or equal to 200,00

Answer is : 5x+4y greater than or equal to 650,000

X less than or equal to 500,000

Y is less than or equal to 200,000

The coordinates of the midpoint of GH¯¯¯¯¯¯¯¯are M(−2, 5) and the coordinates of one endpoint are H(−3, 7) .

The coordinates of the other endpoint are ( __ , __ )

Answers

m = (x1 + x2)/2 , (y1 + y2)/2

(-3 + x) / 2 = -2
-3 + x = -4
x = -4 + 3
x = -1

(7 + y) / 2 = 5
7 + y = 10
y = 10 - 7
y = 3

so the other endpoint is (-1,3)

The coordinate point of the other endpoint, G is (-1, 3)

The given parameters;

the midpoint = M(-2, 5)

one end point = H(-3,7)

To find:

the coordinate point of the other endpoint, G

Let the coordinate point of the other endpoint = G(x, y)

The coordinate point of the other endpoint, G will be calculated using midpoint formula.

[tex]M(-2, 5) =( \frac{x + (-3)}{2} , \frac{y + 7}{2} )\\\\-2 = \frac{x - 3}{2} \ \ \ and \ \ 5 = \frac{y+ 7}{2} \\\\x-3 = 2(-2)\\\\x - 3 = -4\\\\x = -4 + 3\\\\x = -1\\\\y+ 7 = 2(5)\\\\y + 7 = 10\\\\y = 10 -7\\\\y = 3\\\\G= (-1, 3)[/tex]

Thus, the coordinate point of the other endpoint, G is (-1, 3)

Learn more here: https://brainly.com/question/10812174

Oh Plz help - I can't handle this one :)

Answers

a: 5^2*pi = 25pi =~ 78.5375, 10*5*2*pi = 100pi =~ 314.15, 314.15+2(78.5375) =  471.23 cm^2

b: 4*0.5*1.5 = 3, 6*4 = 24, 6*2.5 = 15.5, 15.5*2 + 3*2 + 24 = 61m^2

c: 5.2*3.5 = 18.2, 2.4*3.5 = 8.4, 2.4*5.2 = 12.48, 12.48*2 + 8.4*2 + 18.2*2 = 78.16ft^2

A bit late but hope it helps
a. cynlinder
A = 2 *pi * r^2 + 2 * pi * r * h
A = 2 * pi * 5^2 + 2 * pi * 5 * 10
A = 50 pi + 100 pi
A = 150 * pi cm^2 (Exact Answer)
A = 471 cm^2       (Decimal Approximation)

b.  A(triangles)
A = (1/2) * b * h * 2
A = (1/2) * 4 * 1.5 * 2 
A = 6 m^2
    A(rectangles)
A = L * W
A = 4 * 6 + 2.5 * 6 * 2
A = 24 + 30
A = 54 m^2
Total area = 6 + 54 = 60 m^2

c.  A(rectangular prism)
A = 2LW + 2LH + 2WH
A = 2 * 5.2*2.4 + 2 * 5.2 * 3.5 + 2 * 2.4 * 3.5
A = 78.16 ft^2 

d.  A(rectangular prism)
A = 2LW + 2LH + 2WH
A = 2 *12 * 6 + 2 * 12 * 4.5 + 2 * 6 * 4.5
A = 306 ft^2

e.   You have a large box to wrap and need to know how much wrapping paper will be needed to wrap the box.
The dimensions of the box are  L = 10 in.,  W = 4 in., H = 6 in.

2x+3y=-2 4x+7y=-6 solve for the system of equations

Answers

elliminateion

multiply first equation by -2 and add to 2nd equation


-4x-6y=4
4x+7y=-6 +
0x+1y=-2

y=-2
sub back

2x+3y=-2
2x+3(-2)=-2
2x-6=-2
2x=4
x=2

(x,y)
(2,-2)

Evaluate P(6,2).
A.) 12
B.) 30
C.) 360

Answers

Permutation
[tex]6P2=\frac{6!}{(6-2)!}=\frac{4!\cdot 5 \cdot 6}{4!}=5 \cdot 6=30[/tex]

The value of the [tex]\rm ^6p_2[/tex] is 30.

We have to determine

The value of  P(6,2).

According to the question

The value of the [tex]\rm ^6p_2[/tex] is determined by using permutation.

The probability of selecting an ordered set of 'r' objects from a group of 'n' number of objects.

The order of objects matters in the case of permutation.

The formula to find [tex]\rm ^np_r[/tex] is given by:

[tex]\rm ^np_r = \dfrac{n!}{(n-r)!}[/tex]

Substitute n = 6 and r = 2 in the formula;

[tex]\rm ^np_r = \dfrac{n!}{(n-r)!}\\ \\ \rm ^6p_2 = \dfrac{6!}{(6-2)!}\\ \\ \rm ^6p_2= \dfrac{6 \times 5\times 4 \times 3\times2 \times 1}{4!}\\ \\ ^6p_2= \dfrac{6 \times 5\times 4 \times 3\times2 \times 1}{4\times 3 \times2 \times 1}\\ \\ ^6p_2= 6 \times 5}\\ \\ ^6p_2= 30[/tex]

Hence, The value of the [tex]\rm ^6p_2[/tex] is 30.

To know more about Permutation click the link given below.

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Use the functions h(x) = 2x + 5 and t(x) = 7x − 6 to complete the function operations listed below.

Part A: Find (h + t)(x). Show your work. (3 points)

Part B: Find (h ⋅ t)(x). Show your work. (3 points)

Part C: Find h[t(x)]. Show your work. (4 points)

Answers

We have to use the functions:
h ( x ) = 2 x + 5 and t ( x ) = 7 x - 6
Part A:
( h + t ) ( x ) = ( 2 x + 5 )  + ( 7 x - 6 ) = 9 x - 1
Part B :
( h · t ) ( x ) = ( 2 x + 5 ) · ( 7 x - 6 ) = 14 x² - 12 x + 35 x - 30 =
= 14 x² + 23 x - 30
Part C :
h [ t ( x ) ] = h ( 7 x - 6 ) = 2 · ( 7 x - 6 ) + 5 = 14 x - 12 + 5 =
= 14 x - 7

Suppose there are 10,000 civilizations in the milky way galaxy. if the civilizations were randomly distributed throughout the disk of the galaxy, about how far (on average) would it be to the nearest civilization? (hint: start by finding the area of the milky way's disk, assuming that it is circular and 100,000 light-years in diameter. then find the average area per civilization, and use the distance across this area to estimate the distance between civilizations.)

Answers

The shape of the galaxy is a circular, with diameter 100, 000 lights year, 
so the radius is 50, 000 lights year, thus

 Area of the galaxy is [tex]A(galaxy)= \pi r^{2}= \pi (50,000)^{2}=25*10^{8} \pi [/tex] (lights year square)

The area per civilisation is [tex] \frac{25*10^{8} \pi }{10,000}=25*10^{4} \pi[/tex]

from this result we can assume as a circle of Radius=[tex]25*10^{2}=2500[/tex] lights year.

and assume that each civilisation lives in a planet at the center of these smaller circles.

Check the figure attached.

Planet A, and B are closest in distance. They are each at the center of a circle with radius 2500, and they are tangent. So the distance of A to B is 2500+2500=5000 (light years).


Answer: 5000 light years

(48.9, -53.3) (84.6, -19.9) (-62.6, 81.7) (-14.4 ,-42.6) (71.2, -84.1) (71.2, 76.8 ) state the domain

Answers

The x values are the first one (and correlate to the domain), so we just have to find the smallest and largest number, which are -62.6 and 84.6 by just looking through the data

I want to make sure of my answer is right?

Answers

20 2/7 = 142/7 -20/35

find common denominator ( in this case it is 35)

35/7 =5 , multiply 142*5=710, 7*5 =35

142/7 = 710/35

710/35 - 20/35  = 710-20 = 690

690/35 now simplify that fraction

690/35 = 19.714

19*35 = 665

690-665=25

19 25/35

25/35 can reduce to 5/7

answer = 19 5/7


One number is 2 more than 3 times another. their sum is 22. find the numbers. 8, 14

Answers

x + y = 22
x = 3y + 2

3y + 2 + y = 22
4y + 2 = 22
4y = 22 - 2
4y = 20
y = 20/4
y = 5

x = 3y + 2 = 3(5) + 2 = 15 + 2 = 17

ur numbers are 5 and 17
Let the number be x.

Hence the other number is 2 + 3*x.

Their sum is 22;

x + 2 + 3x = 22

4x + 2 = 22

4x = 20

 Therefore, x = 5

[ 2 + 3x = 2 + 3*5 = 2+15 = 17 ]

Hence the two numbers are, 5 and 17. 

33.
In quadrilateral PQRS, the coordinates are P(0, 0), Q(a + c, 0), R(2a + c, b), and S(a, b). How can you use coordinate geometry to show that the diagonals are perpendicular?

Apply the Distance Formula to show that opposite sides to show they  are congruent.

Find the slopes of  their product is 1.

Apply the Distance Formula to show that opposite sides to show sides  are congruent.

Find the slopes and show that their product is -1.

Answers

By definition, perpendicular line are two lines that intersect at right angles. In other words, the angle made by two lines should be 90°. Therefore, the use of distance formula does not help because it only tells you if the sides are equal. It does not tell you about the intercepted angle.

A technique that can help you to know if two straight lines are perpendicular is is you find their slopes. Let's say the slope of line 1 is m1 and the slope of line 2 is m2. If m1*m2 yields a product of -1, then the lines are perpendicular. This is because if m1 is the negative reciprocal of m2, the lines are perpendicular. But if m1=m2, the lines are parallel, meaning they don't intersect at all.

Therefore, the answer is: Find the slopes and show that their product is -1.

Answer: The answer is (d) Find the slopes and show that their product is -1.

Step-by-step explanation:  Given in the question and shown in the attached figure that the vertices of the quadrilateral PQRS are P(0, 0), Q(a + c, 0), R(2a + c, b), and S(a, b). We are asked to use geometry to show that the diagonals PR and QS are perpendicular to each other.

We know that the two lines are perpendicular if the product of their slopes is --1.

So, first we will find the slopes of PR and QS, multiply them, and check the value.If the value is -1, then the diagonals are perpendicular to each other.

That is, if 'm' and 'p' are the slopes of the diagonals PR and QS, and if m × p = -1, then the two diagonals are perpendicular.

Thus, the correct answer is (d).

Part B: Find an irrational number that is between 9.5 and 9.7. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth

Answers

irrational number between 9.5 and 9.7...

9.678937... (never ending)...it is irrational because it cannot be made into a fraction because it is infinite.

the decimal approximation to the nearest hundredth is : 9.68

Wich of the operations ,+,-,x➗are inverse o each other

Answers

The correct answers are as follows: Adding (+) and subtracting (-) are inverse to each other. And then multiplication (x) is inverse to division (➗)

What are the explicit equation and domain for an arithmetic sequence with a first term of 5 and a second term of 2?
an = 5 − 2(n − 1); all integers where n ≥ 1
an = 5 − 2(n − 1); all integers where n ≥ 0
an = 5 − 3(n − 1); all integers where n ≥ 1
an = 5 − 3(n − 1); all integers where n ≥ 0

Answers

The easiest way to answer this is to try all choices, plug in values for the 1st term and 2nd term then check if the answer matches with 5 and 2. (n = 1 and n = 2)

We know that n starts with 1 because that is our 1st term, we do not have 0th term, therefore that leaves us with 2 choices.

Choice 1: an = 5 − 2(n − 1); all integers where n ≥ 1

n = 1

a1 = 5 – 2 (1 – 1) = 5 – 2 (0)

a1 = 5

 

n = 2

a2 = 5 – 2 (2 – 1) = 5 – 2 (1)

a2 = 3   (FALSE!)

 

Choice 2: an = 5 − 3(n − 1); all integers where n ≥ 1

n = 1

a1 = 5 – 3 (1 – 1) = 5 – 3 (0)

a1 = 5

 

n = 2

a2 = 5 – 3 (2 – 1) = 5 – 3 (1)

a2 = 2   (TRUE)

 

Therefore the correct answer is:

an = 5 − 3(n − 1); all integers where n ≥ 1

Answer:

A is INCORRECT

Step-by-step explanation:

Just finished the FLVS test

A textile corporation buys equivalent with an initial purchase price of $750000. It is estimated its useful life will be 3 years and at that time it's value will be$75000. The total depreciation is depreciation is divided equally among the three years. What is the total amount of depreciation declared each year?

Answers

First,we have to find the depreciation, which is 750000-75000=675000. Dividing that by 3 (for 3 years), we get 225000 dollars per year

Let $f$ be a function such that $f(x+y) = x + f(y)$ for any two real numbers $x$ and $y$. if $f(0) = 2$, then what is $f(2012)?$

Answers

Given:

f(0) = 2

So first of all, we let x = 2012, y = 0: 


Then, F(2012) = 2012 + f(0)

Since f(0) = 2, then f(2012) = 2012 + 2 = 2014.

To add, the process that relates an input to an output is called a function.

 

 

There are always three main parts of a function, namely:

Input

The Relationship

The Output

 

The classic way of writing a function is "f(x) = ... ".

 

What goes into the function is put inside parentheses () after the name of the function: So, f(x) shows us the function is called "f", and "x" goes in.

 

 

What a function does with the input can be usually seen as:

 

f(x) = x2 reveals to us that function "f" takes "x" and squares it.

The sum of the first and third of three consecutive even integers is 136. find the three even integers.

Answers

Based on your problem, you only have one given. So, you can't make an equation for this because there are not limits to the equation. The only thing that you know is that the three numbers are consecutive even integers. My way of solution for this is trial-and-error. However, it's really quite easy. 

For example: 42 + 46 = 88. I have to increase the numbers more to reach 136. Suppose: 82 + 86 = 168. That exceeded 136. So, it must be between 46 and 82. Suppose again: 66 + 70 = 136. Therefore, the sequence of the consecutive even integers are 66, 68, and 70

The sum of the first and third even integers is 136. By setting up an equation and solving for x, where x is the first integer, we determine that the consecutive even integers are 66, 68, and 70.

The problem asks us to find three consecutive even integers where the sum of the first and third integer is 136. Let's denote the first integer as x, the second integer as x + 2, and the third integer as x + 4 (since consecutive even numbers have a difference of 2). To solve for the values, we set up the equation x + (x + 4) = 136. Simplifying, we get 2x + 4 = 136. Subtracting 4 from both sides gives us 2x = 132. Dividing both sides by 2, we find out that x = 66. So, the first integer is 66, the second integer is 66 + 2 = 68, and the third integer is 66 + 4 = 70.

Therefore, the three consecutive even integers are 66, 68, and 70.

how would you solve it. I know the answer already. my problem working it out.
4b/2 + 3 = 11 + b.

Answers

-3 to both sides
4b/2 = 8 +b

multiply both sides by two
(4b/2 = 8 + b) X2    =    4b = 16+2b

subtract 2b from both sides
2b = 16

divide both sides by 2
b=8
How to solve an equation.

1. Simplify any parentheses using the distributive property if needed.
2. Combine like terms if needed.
3. Use the addition principle of equality if needed.
4. Use the multiplication or division principle of equality if needed.

[tex]\frac{4b}{2} + 3 = 11 + b[/tex]
2b + 3 = 11 + b
b + 3 = 11 <-- Subtract b from each side
b = 8 <-- Subtract 3 from each side

So, b is equal to 8.

A quadratic equation is shown below: 3x2 − 15x + 20 = 0 Part A: Describe the solution(s) to the equation by just determining the radicand. Show your work. (5 points) Part B: Solve 3x2 + 5x − 8 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used. (5 points)

Answers

Answer:

[tex]\text{The roots of }3x^2+5x-8=0\text{ are }x=1,\frac{-8}{3}[/tex]

Step-by-step explanation:

[tex]\text{Part A: Given a quadratic equation }3x^2-15x+20=0[/tex]  

[tex]\text{Comparing above equation with }ax^2+bx+c=0[/tex]  

a=3, b=-15, c=20

Discriminant can be calculated as

[tex]D=b^2-4ac[/tex]

[tex]D=(-15)^2-4(3)(20)=225-240=-15<[/tex]

The roots are imaginary

The solution is

[tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]

[tex]x=\frac{-(-15)\pm \sqrt{-15}}{2(3)}=\frac{15\pm\sqrt{15}i}{6}[/tex]

The roots are not real i.e these are imaginary    

[tex]\text{Part B: Given a quadratic equation }3x^2+5x-8=0[/tex]  

[tex]\text{Comparing above equation with }ax^2+bx+c=0[/tex]  

a=3, b=5, c=-8

Discriminant can be calculated as

[tex]D=b^2-4ac[/tex]

[tex]D=(5)^2-4(3)(-8)=25+96=121>0[/tex]

The roots are real

By quadratic formula method

The solution is

[tex]x=\frac{-b\pm\sqrt{D}}{2a}[/tex]

[tex]x=\frac{-5)\pm \sqrt{121}}{2(3)}=\frac{-5\pm 11}{6}[/tex]

[tex]x=1,\frac{-8}{3}[/tex]

which are required roots.

I choose this method because I can get the solutions directly by substituting the values in formula, and I don't have to guess the possible solutions.

WILL MARK BRAINIEST!!

e) Isabel’s competition,who sells books and no other products, charges a shipping fee of $1.99 per book plus a fixed fee of $3 for each order. Let x be the number of books purchased and f(x) be the price of shipping one order of books. Write a function that expresses f(x) and explain the value of f(x) for x = 0. (5 points)

f) Graph the function in part e. (10 points)

g) Is your graph discrete or continuous? Explain. (5 points)

Answers

Answer:

Isabel’s competition,who sells books and no other products, charges a shipping fee of $1.99 per book plus a fixed fee of $3 for each order. Let x be the number of books purchased and f(x) be the price of shipping one order of books. Write a function that expresses f(x) and explain the value of f(x) for x = 0. (5 points)

The price of the shipping ( f(x)) is equal to the cost of the fixed fee per order (3) plus the product of the number of books ordered (x) and the cost of the fee per book (1.99).

f(x)= 1.99x +3

for f(0) = 1.99(0)+3

f(0) =3

For F(0) there is a order of 0 books.  The price is only for the shippping.

Step-by-step explanation:thats it buddy

A function that expresses f(x) is f(x) = 1.99x + 3. Graph is continuous.

What is continuous function?

A continuous function is a function that does not have discontinuities that means any unexpected changes in value. A function is continuous if we can ensure arbitrarily small changes by restricting enough minor changes in its input.

For this case we have the following data:

A shipping fee of $ 1.99 is charged for each book sold.

There is a fixed fee of 3 dollars per order.

We must express the total cost by a function of the form .

If x represents the number of books sold, we have an expression of the form:

f(x) = 1.99x + 3

That is, according to the number of books sold, represented by x, a different value will be obtained for the total cost given by function f (x). For example:

If 3 books are sold, we have:

f(x) = 1.99x + 3

f(0) = 1.99(0) + 3

f(0) = 3

Thus, the value of f(x) for x = 0 is f(0) = 3.

The graph is continuous. Continuous is something that can and must be able to be broken down into fractions and decimals. Discrete means something that cannot be broken down into fractions or decimals.

Find out more information about continuous function here:

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Use the arc length formula to find the length of the curve y = 5x − 4, −1 ≤ x ≤ 3. check your answer by noting that the curve is a line segment and calculating its length by the distance formula

Answers

The given curve is y = 5x -4,  -1 ≤ x ≤ 3.

The length of the arc is computed from the formula
[tex]S= \int_{-1}^{3} \,\sqrt{1+ (\frac{dy}{dx} )^{2}} \, dx[/tex]

The derivative is 
y' = 5

Therefore
[tex]S = \int_{-1}^{3} \sqrt{1+25} \, dx =\sqrt{26}*(3-(-1))=20.396[/tex]

Note that 
x = -1 +> y = 5(-1) - 4 = -9
x = 3 => y = 5(3) - 4 = 11
The distance between the points (-1, -9) and (3, 11) from the distance formula is
D = √[(3-(-1))² + (11-(-9))²] = √(16+400) = 20.396
This answer agrees with that obtained by integration.

Answer: 20.396
Obtained by integration and verified by the distance formula.

The arc length is 4√26, which is confirmed by the distance formula for a line segment yielding approximately 20.396 units.

To find the length of the curve y = 5x − 4 over the interval −1 ≤ x ≤ 3, we can use the arc length formula for a function y = f(x). The formula for the arc length L is given as:

L = ∫ab√(1 + (dy/dx)2) dx, where dy/dx is the derivative of y with respect to x.

1. Calculate the derivative dy/dx:

y = 5x − 4, so dy/dx = 5.

2. Substitute the derivative into the arc length formula:

L = ∫-13√(1 + 5²) dx = ∫-13√26 dx.

3. Integrate the function:

L = √26 ∫-13 dx = √26 [x]-13 = √26 (3 − (-1)) = 4√26 ≈ 20.396.

To verify this using the distance formula (since the curve is a straight line), we take the points (−1,−9) and (3,11) from the given curve:

4. Apply the distance formula:

Distance = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) = (-1, -9) and (x2, y2) = (3, 11).

Therefore,

Distance = √((3 - (-1))² + (11 - (-9))²) = √((4)² + (20)²) = √(16 + 400) = √416 = 4√26 ≈ 20.396.

Thus, the arc length calculated using the formula and the distance formula are consistent.

The arc length is 4√26, which is confirmed by the distance formula for a line segment yielding approximately 20.396 units.

You have a cone with a radius of 4 ft and a height of 12 ft. What is the height of the triangle formed by a perpendicular cross-section through the cone’s center?

Answers

check the picture.

ABC is the triangle formed by a perpendicular cross-section through the cone's center.

The height OC of this triangle, is the height of the cone.

So the height is equal to 12 ft.


Answer: 12 ft.
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