Find the volume of the tetrahedron having the given vertices. (3, −1, 1), (5, −2, 4), (1, 1, 1), (0, 0, 1)

Answers

Answer 1

Given four vertices of a tetrahedron, we need to find its volume.
A(3,-1,1), B(5,-2,4), C(1,1,1), D(0,0,1)

The simplest method is to use vectors.

Here, we calculate the volume of a parallelepiped defined by vectors AB, AC, AD. The volume of a tetrahedron is one-sixth of the volume of the parallelepiped, V, given by AB.(ACxAD).
Here are the steps:
1. Choose a vertex, say A. Calculate vectors AB, AC and AD
AB=(5-2,4)-(3,-1,1) = <2, -1, 3>
AC=(1,1,1)-(3,-1,1) = <-2, 2, 0>
AD=(0,0,1)-(3,-1,1) = <-3, 1, 0>
2. Calculate vector cross product ACxAD = <-2,2,0>x<-3,1,0>
i j k
-2 -2 0
-3, 1, 0
=<0-0, 0-0, -2-(-6)>
= <0, 0, 4>
3. Calculate volume of parallelepiped, equal to the dot product AB.(ACxAD), ignore sign, which is arbitrary.
V = |<2, -1, 3>.<0,0,4>| = |12| = 12
4. Calculate volume of tetrahedron, which is one-sixth of volume of parallelepiped
V(tetrahedron) = V/6= 12/6 = 2

Related Questions

library library currently has 125 audiobook the library plan purchase the same number of books each month after 4 months the library has 20 more audiobook which equation can be used to determine t and a total number of audio books the library will have after and months

Answers

125=(4t)+20 I'm not sure about the variables as they aren't defined

Find the coefficient of x 4 in the taylor series expansion centered at the origin for the function f(x) = 8 ln(9 + 3x 2 ).

Answers

Try this option.
Note, that it is the full solution, it is possible to make it shorter.

Final answer:

The coefficient of x4 in the Taylor series expansion of the given function is 0.

Explanation:

To find the coefficient of x4 in the Taylor series expansion of the function f(x) = 8 ln(9 + 3x2), we need to compute the fourth derivative of f(x) and evaluate it at x = 0.

The Taylor series expansion for a function centered at the origin is given by:

f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + f''''(0)x4/4! + ...

In this case, we have:

f'''(x) = 48x/(9 + 3x2)

Evaluating f'''(x) at x = 0 gives:

f'''(0) = 0/9 = 0

Therefore, the coefficient of x4 in the Taylor series expansion centered at the origin for the function f(x) = 8 ln(9 + 3x2) is 0.

Eliminate the parameter. x = one divided by two t, y = 3t^3 - 1

Answers

Your answer to this willl be undefined because you do not have enough information to complete the question

Answer:

y = 24x3 - 1

Step-by-step explanation:

None needed YW

Martin put 8 pounds of grapes into a bowl. Four friends each took 2/3 pounds from the bowl. What fraction represents the total amount of grapes removed from the bowl ?

Answers

There are 8 pounds of grapes. Four people each take 2/3 out. We can find out how much was taken out by multiplying these four people by the amount they took out, 2/3:

[tex]\sf 4\times\dfrac{2}{3}=\dfrac{8}{3}[/tex]

We can convert this improper fraction to a mixed number. Three goes into eight two times. 3 * 2 = 6. 8 - 6 = 2. So we have 2 and two thirds leftover, or:

[tex]\sf 2\dfrac{2}{3}[/tex]

So the total amount of grapes that was removed from the bowl is [tex]\sf\dfrac{8}{3}[/tex] or [tex]\sf 2\dfrac{2}{3}[/tex] pounds.

Answer:

-8/3

Step-by-step explanation:

i took the test on edg

thank youuuuuuuuuuuuuu

Answers

the perimeter, as you already know, is the sum of all the sides.

14)

[tex]\bf (x+3)+(2x+4)+(y)=\stackrel{perimeter}{6x}\implies 3x+7+y=6x \\\\\\ 7+y=3x\implies y=3x-1[/tex]

15)

a rectangle has twin widths and twin lengths, therefore

[tex]\bf \stackrel{width}{(3x-3)}+\stackrel{width}{(3x-3)}+\stackrel{length}{(y)}+\stackrel{length}{(y)}=\stackrel{perimeter}{10x+6} \\\\\\ 6x-6+2y=10x+6\implies -6+2y=4x+6\implies 2y=4x+12 \\\\\\ y=\cfrac{4x+12}{2}\implies y=\cfrac{4x}{2}+\cfrac{12}{2}\implies y=2x+6[/tex]

Which are the solutions of x2 = –5x + 8?

Answers

The answer is A
hope this helps

Answer:

The solutions are 1.275 and -6.275

Step-by-step explanation:

Given

x^2 = –5x + 8

First, put all terms in the left side of the equal sign, as follows:

x^2 + 5x - 8 = 0

Then, use the quadratic formula with a = 1 (the coefficient of the quadratic term), b = 5 (the coefficient of the linear term) and c = -8 (the independent term).

[tex]x = \frac{-b \pm \sqrt{b^2 - 4(a)(c)}}{2(a)}[/tex]

[tex]x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-8)}}{2(1)}[/tex]

[tex]x = \frac{-5 \pm \sqrt{57}}{2}[/tex]

[tex]x_1 = \frac{-5 + \sqrt{57}}{2}[/tex]

[tex]x_1 = \frac{-5 + 7.55}{2}[/tex]

[tex]x_1 = 1.275[/tex]

[tex]x_2 = \frac{-5 - \sqrt{57}}{2}[/tex]

[tex]x_2 = \frac{-5 - 7.55}{2}[/tex]

[tex]x_2 = -6.275 [/tex]

A surveyor holds a laser range finder 3 feet above the ground and determines that the range finder is 89 feet from a building and 115 feet from the top of the building. What is the height h of the building? Round your answer to the nearest tenth.

Answers

The distance to the top of the building above the range finder can be found from the Pythagorean theorem.
.. d = √(115² -89²) ≈ 72.8 . . . . feet

Adding the height of the range finder, we find the height of the building to be
.. h = d+3 = 75.8 ft

How many positive integers between 100 and 999 inclusive are divisible by three or four?

Answers

999 - 100 + 1 = 900
There are 900 numbers between 100 and 999 (inclusive)

We group the numbers into groups of 3s. So we find one number divisible by 3 in every group.
Number of numbers that can be divided by 3 = 900 ÷ 3 = 300
Number of numbers that can be divided by 3 = 300

We group the numbers into groups of 4s. So we find one number divisible by 4 in every group.
Number of numbers that can divided by 4 = 900 ÷ 4 = 225
Number of numbers that can divided by 4 = 225
Among the group of 4s, there is 1/3 that are already inclusive in divisible by 3.
225 ÷ 3 = 75
Number of numbers that can divided by 4 = 225 - 75 = 150

Total numbers of positive integers divisible by three and fours = 300 + 150 = 450

Final answer:

The total number of positive integers between 100 and 999 inclusive that are divisible by 3 or 4 is 450.

Explanation:

The problem is looking for positive integers between 100 and 999 inclusive, that are divisible by three or four. First, we find out how many numbers between 100 and 999 are divisible by 3 by using the formula: (Last Numbers - First Number) divided by the number you are dividing + 1. Hence, [tex](999 - 102) / 3 + 1 = 300.[/tex] Similarly, for numbers divisible by 4: (996 - 100) / 4 + 1 = 225. However, we must subtract out the numbers that are counted twice because they are divisible by both 3 and 4 (i.e., divisible by 12). To find this, we do [tex](996 - 108) / 12 + 1 = 75.[/tex]

In conclusion, we add the numbers divisible by 3 and those divisible by 4 and subtract the ones that are divisible by both (12). Hence, the total number of positive integers between 100 and 999 inclusive that are divisible by 3 or 4 is [tex]300 + 225 - 75 = 450.[/tex]

Learn more about Integers divisible by 3 or 4 here:

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Point T' has coordinates (-3, 4). If it was translated up 3 units, what were the coordinates of its pre-image?

(0, 4)
(-3, 7)
(-3, 1)
(-6, 4)

Answers

Translation refers to a rigid motion or transformation where you move an object from one location to another.
Translations are base on the rule (x,y) -- (x+a, y+b) where the words right and left represent the change in the x-axis, and the words up and down the change in the y-axis.

In this exercise it is asked you to find the coordinates of the preimage of point T', which has the coordinates (-3,4) and is the effect of a translation of 3 units up. As previous said, the words up and down represent a change on the y-axis, meaning that the y-coordinate is the one that should be changed.

If the translation from the preimage to the image was 3 units up, then the preimage can be found by subtracting three units.
Pre-image: (-3,4-3)= (-3,1)

The point that represents the coordinates of the pre-image is (-3,1).

Answer: the answer is (-3, 1)

yes I know I am late :)

What is the circumference of a circle with a radius of 5.4 inches? Enter your answer as a decimal in the box. Use 3.14 for pi. Round your answer to the nearest tenth. in.

Answers

Hi there! The formula for circumference is π * diameter. Diameter is the length across the circle and twice the length of the radius.The radius is 5.4, so we will multiply that by 2 to get 10.8. The diameter is 10.8 inches. Now, multiply that numebr by 3.14 to find the circumference. 10.8 * 3.14 is 33.912 or 33.9 when rounded to the nearest tenth. There. The circumference is approx. 33.9 inches.

Answer:

33.9

Step-by-step explanation

I got 100% on my test and just wanted everyone to know that it is correct.

In a normal distribution, most numbers in the data set fall within one standard deviation of the mean. True or False

Answers

Answer:

True

Step-by-step explanation:

Plzz plzzzzz help !!

Answers

The answer is (B&C) hope it helps and have a great day!

!!!!!GIVING BRAINLIEST!!!!!

Select the graph for the solution of the open sentence. Click until the correct graph appears. 3|x| > 6

Answers

Shown below is the graph on the x-y plane.

Your equation has only one variable, so a graph on the number line is appropriate. It would extend outward from ±2, where the points would be indicated by open circles.

Answer:

Given equation,

3|x| > 6 -----(1)

∵ [tex]\frac{1}{3}>0[/tex]

Thus, multiplying both sides of inequality (1) by [tex]\frac{1}{3}[/tex]

We get,

[tex]|x| > 2[/tex]

⇒ 2 < x or 2 < - x

⇒ 2 < x or -2 > x (when a > b where a > 0, b > 0, If c < 0 ⇒ ac < bc),

⇒ (2, ∞ ) ∪ (-∞, -2)

By plotting the above interval in the number line we can get the graph of the given equation ( shown below )

What rigid transformation would map ABC to EDC?

Answers

I believe it would be a rotation around line c, the last option

Answer: D is the answer

Step-by-step explanation:

D^2+d-30/d^2+3d-40 + d^2+14d+48/d^2-2d-48
A. D^2+14d+16/(d+8)(d-8)
B. 2d^2+14d+16/(d+8)(d-8)
C.2d^2+15d+18/2d^2+d-88
D.2d^2+15d+18/(d+8)(d-8)

Answers

I think D.2d^2+15d+18/(d+8)(d-8)

Answer: option B.

   2d^2 + 14d + 16
----------------------------
        (d+8)(d-8)

Explanation:

The question is:

[tex] \frac{d^2+d-30}{d^2+3d-40} + \frac{d^2+14d+48}{d^2-2d-48} [/tex]

1) Start factoring all the polynomials to rewrite the fractions.

2) d^2 + d - 30 = (d + 6)(d - 6)

3) d^2 + 3d - 40 = (d + 8)(d - 5)

4) d^2 + 14d + 48 = (d + 6) (d + 8)

5) d^2 - 2d - 48 = (d - 8)(d + 6)

6) rewrite the fractions:

(d+6)(d-5)         (d+8)(d+6)
----------------- + -----------------
(d+8)(d-5)          (d-8)(d+6)

7) simplify the fractions cancelling the factors that are equal in the numerator and the denominator:

d+6     d+8
-------- + -------
d+8       d-8

8) take least common denominatior: (d+8)(d-8), and sum the fractions:

   (d-8)(d+6) + (d+8)^2
--------------------------------
        (d+8)(d-8)

9) expand the parenthesis in the numerator and combine like terms:

d^2 - 2d - 48 + d^2 + 16d + 64
------------------------------------------- =
               (d+8)(d-8)

        2d^2 + 14d + 16
=   ----------------------------
            (d+8)(d-8)

And that is the option B.

Can someone help quickly plz

Answers

we have that

using a graph tool
see the attached figure

the answer is the first option
y<=3x
y>-2x-1

A rectangular window is 1 meter high and 2 meters wide. What is the perimeter of the window in meters? What is it in centimeters?

Answers

The perimeter is the total length of all sides of the shape.

It is 1 meter + 1 meter + 2 meter + 2 meter = 6 meters

1 meter = 100 centimeter

6(100) = 600 centimeter

Hope this helps! :)

A football stadium holds 52,000 fans. A college student is doing research and determines that on any given game day, the home team has five times as many fans as the visiting team. In order to help the student in his research, he represents the number of home team tickets as H and the visiting team’s tickets as V.

Answers

You probably are interested in setting up system of equations to find the number of fans for Home Team and Visiting Team.

Based on this data, we can find the number of number of fans of each team.

Following data has been provided:

1) Total number of fans stadium can hold = 52000
So on a given day if stadium is full,
H + V = 52000

2) Home team has 5 times as many fans as the visiting team.
So,
H = 5V

Using the value of H in first equation, we get:

5V + V = 52000
6V = 52000
V = 8667 fans

H + V = 52000
⇒
H = 52000 - V = 52000 - 8667 = 43333 fans

So,
there will be 43,333 fans of home team and 8667 fans of visiting team.

ava pours 3 gallons of paint equally into 5 cans

Answers

Ok what is the question

Each can will have 3/5 gallon of paint.

You want to have $31,664 in cash to buy a car in 4 years. you expect to earn 11.5% per year on your savings. how much do you need to deposit today if this is the only money you save for this purpose?

Answers

You would need to deposit approximately $20,701.27 today if this is the only money you save for the purpose of buying a car in 4 years, considering an expected annual interest rate of 11.5%.

We have,

To determine how much you need to deposit today, we can use the concept of present value (PV).

The present value represents the current value of a future cash flow, accounting for the time value of money and the expected interest rate.

In this case, you want to have $31,664 in cash in 4 years, and you expect to earn an annual interest rate of 11.5%.

Let's calculate the present value using the following formula:

[tex]PV = FV / (1 + r)^n[/tex]

where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods.

Plugging in the values, we have:

[tex]PV = $31,664 / (1 + 0.115)^4[/tex]

Calculating this equation, we find:

[tex]PV = $31,664 / (1.115)^4[/tex]

= $31,664 / 1.530749

PV ≈ $20,701.27

Therefore,

Learn more about expressions here:

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Josef says that the range for the car data is greater than the range for the minivan data because the box in the box plot for the car data is wider. Which explains Josef’s error?

Answers

Josef confused the range and the interquartile range.Josef confused the range and the median.Josef should have compared the medians and minimum values.Josef should have compared the medians and maximum values.

Choice A

Answer:

its a ik it

Step-by-step explanation:

I need this equation solved with the work shown please

Answers

X = -2 1/2 because -2 1/2 to the power of to is -5 and -5 + -4 = -9 and -9 + 9 =0

SO THE ANSWER IS 2 1/2

△ABC is reflected to form △A′B′C′ . The vertices of △ABC are A(−7, 1) , B(−5, −3) , and C(−3, 2) . The vertices of △A′B′C′ are A′(−7, −1) , B′(−5, 3) , and C′(−3, −2) . Which reflection results in the transformation of △ABC to △A′B′C′ ?

Answers

the vertices of the preimage and final image are as follows
A(−7, 1) ---> A′(−7, −1)
B(−5, −3) ---> B′(−5, 3)
C(−3, 2) ---> C′(−3, −2)
it can be noted that whilst the x coordinates remain the same in the final image, y coordinate values, the sign changes.
eg: A (x = -7) and A' (x = -7)
therefore x coordinates haven't changed the value nor sign
A (y = 1) and A' (y = -1)
y coordinates have been multiplied by -1 hence changing the sign.
the reflection goes as (x,y) = (x,-y)
this is a reflection over the x axis.

what is the variable in this algebraic expression 8+6, 4d-2, 7r-15=6, 11+14=25

Answers

For this case we have that by definition, a variable is a symbol that can be replaced or that takes a numerical value in an equation or expression, mathematical in general.

Then we have the following options:

A) [tex]8 + 6[/tex]

It is a sum of two real numbers. There are no variables.

B)[tex]4d-2[/tex]

In this case the variable is "d"

C) [tex]7r-15 = 6[/tex]

In this equation, the variable is represented by "r"

D) [tex]11 + 14 = 25[/tex]

It is a sum of two real numbers. There are no variables.

Answer:

A) There are no variables

B) "d"

C) "r"

D) There are no variables

Which expression is equivalent to (2mn)^4/6m^-4n^-2? Assume m=/0 n=/0.

Answers

Answer = 8m⁸ n⁶/3

EXPLANATION

(2mn)^4/6m^-4n^-2

⇒(2mn)⁴/6m⁻⁴n⁻²

= 2⁴ · m⁴ · n⁴ / 6 · m⁻⁴ · n⁻²

= [tex] \frac{2 ^{4} }{6} [/tex] · m⁴⁻⁽⁻⁴⁾ · n⁴⁻⁽⁻²⁾

= [tex] \frac{16}{6} [/tex] · m⁴⁺⁴ · n⁴⁺²

= [tex] \frac{8}{3} [/tex] · m⁸ · n⁶

= 8m⁸ n⁶/3
Same as (8m^8n^6)/3

True: If it is a function
False: If it is not a function

{(2, 3), (4, 5), (4, 3), (3, 4)}

Answers

Answer: false.

Explanation:

The given relation does not fit the definition of function.

A function is a relationship that states unambigously the image (output) of the input (independent) variable.

In the set of ordered pairs it is true form the input-values 2, and 3, because their images are 3 and 4, respectively.

But, could you tell the image of 4?. It may be 5 or 3. That is an ambiguity. That is why the given relationship is not a function.

A function is defined as a set of ordered pairs no two of which have the same first values or first component.

This means if more than one ordered pairs will have the same first component, the relation will not be a function.

In the given relation, the first component of 2nd and 3rd ordered pair is the same, so the relation {(2, 3), (4, 5), (4, 3), (3, 4)} is not a function.

So the Answer is False.

how do you reduce 4/8?

Answers

Just think of 4/8. Four is half of 8, right? therefore, it is 1/2. 4/8 reduced is 1/2

you divide by 2 on each side!!
so you get 2/4 which is equal to 1/2 which is your answer!!
Hope this helps!!

I have created a triangular garden such that the largest side is 8m less than twice the smallest and the medium side is 12m larger than the smallest side. If the total perimeter of the garden is 104m, what are the lengths of the three sides?

Answers

The smallest side is 25 m, the medium side is 37 m, and the largest side is 42 m.

each picnic table seats 6 people how many picnic tables are needed to seat 24 people?

Answers

Final answer:

To seat 24 people at picnic tables that each seat 6, you divide 24 by 6, which equals 4 tables needed.

Explanation:

The question is asking how many picnic tables are needed to seat 24 people, if each table seats 6 people. This is a division problem where we need to divide the total number of people by the number of seats at a single table. You divide 24 (the total number of people) by 6 (the number of seats at one table) to find out the number of tables needed.

Here is the calculation: 24 \/ 6 = 4. Therefore, you would need 4 picnic tables to seat 24 people, assuming each person takes one seat and that there are no empty seats.

if the height of a rectangular prism is doubled then the surface area of the prism is and I need work to PLEASE HELP ME

Answers

The surface area of a rectangular prism (S) is given in terms of length L, width W, and height H by
.. S = 2(LW +H(L +W))

If you double the height, now you have new surface area S'
.. S' = 2(LW +2H(L +W)) = 2(LW +H(L +W)) +2H(L +W)
.. = S +2H(L +W)

This is larger than the original surface area S, but is not as much as 2S.

Selection C is appropriate.

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