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A plane with equation (x/a) + (y/b) + (z/c) = 1, where a,b,c > 0 together with the positive coordinate planes form a tetrahedron of volume V = (1/6)abc. Find the plane that minimizes V if the plane is constrained to pass through the point P(2,1,1).
Answer:
The plane is x/6 + y/3 + z/3 = 1
Step-by-step explanation:
Given
Equation: (x/a) + (y/b) + (z/c) = 1 where a,b,c > 0
Minimise, V = (1/6) abc subject to
the constraint g = 2/a + 1/b + 1/c = 1
First, we need to expand V
V = (abc)/6
Possible combinations of V taking 2 constraints at a time; we have
(ab)/6, (ac)/6 and (bc)/6
Applying Lagrange Multipliers on the possible combinations of V, we have:
∇V = λ∇g
This gives
<bc/6, ac/6, ab/6> = λ<-2/a², -1/b², -1/c²>
If we equate components on both sides, we get:
(a²)bc/12 = -λ = a(b²)c/6 = ab(c²)/6
Solving for a, b and c;
First, let's equate:
(a²)bc/12 = a(b²)c/6 -- divide through by abc, we have
a/12 = b/6 --- multiply through by 12
12 * a/12 = 12 * b/6
a = 2 * b
a = 2b
Then, let's equate:
(a²)bc/12 = ab(c²)/6 -- divide through by abc, we have
a/12 = c/6 --- multiply through by 12
12 * a/12 = 12 * c/6
a = 2 * c
a = 2c
Lastly, we equate:
a(b²)c/6 = ab(c²)/6 -- divide through by abc, we have
b/6 = c/6 --- multiply through by 6
6 * b/6 = 6 * c/6
b = 2
Writing these three results, we have
a = 2b; a = 2c and b = c
Recalling the constraints;
g = 2/a + 1/b + 1/c = 1
By substituton, as have
2/(2c) + 1/c + 1/c = 1
1/c + 1/c + 1/c = 1
3/c = 1
c * 1 = 3
c = 3
Since a = 2c;
So, a = 2 * 3
a = 6
Similarly, b = c
So, b = 3
So, the plane: (x/a)+(y/b)+(z/c)=1;
By substituton, we have
x/6 + y/3 + z/3 = 1
Hence, the plane
So the plane is x/6 + y/3 + z/3 = 1
The equation of the plane that minimizes the volume of the tetrahedron and passes through the point (2,1,1) is x + z = 1.
Explanation:To find the plane that minimizes the volume of the tetrahedron, we need to find the equation of the plane that passes through the point P=(2,1,1) and has coefficients a, b, and c. We can substitute the coordinates of the point into the equation of the plane and solve for the a, b, and c.
Substituting the coordinates of P into the equation of the plane gives 2a + b + c = 1. Since a, b, and c are all greater than 0, we can set a = 1, b = 0, and c = 1.
Therefore, the equation of the plane that minimizes the volume of the tetrahedron is x + z = 1.
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I NEED HELP OBN THIS BADLY WILL GIVE YOU CROWN
A jug contained 3 1/4 pints of milk at the start of a baking class. At the end of the class, only 3 fluid ounces were left.
How many fluid ounces of milk were used during the class?
10 fl oz
23 fl oz
49 fl oz
101 fl oz
GEOMETRY! PLEASE SHOW YOUR WORK.
Given:
The base of the given triangle has two part:
Left( left side of the hypotenuse) = 25
Right ( right side of the hypotenuse) = x
The altitude (h) = 60
To find the value of x.
Formula:
By Altitude rule we know that, the altitude of a triangle is mean proportional between the right and left part of the hypotenuse,
[tex]\frac{left}{altitude} =\frac{altitude}{right}[/tex]
Now,
Putting,
left = 25, altitude = 60 and right = x we get,
[tex]\frac{25}{60} = \frac{60}{x}[/tex]
or, [tex](25)(x)=(60)(60)[/tex] [ by cross multiplication]
or, [tex]x= \frac{3600}{25}[/tex]
or, [tex]x=144[/tex]
Hence,
The value of x is 144.
Answer:
x = 144
Step-by-step explanation:
Height Theorem
60² = 25·x
3600 = 25x
x = 3600 : 25
x = 144
If a toy helicopter is moving at 12 m/s, how much energy does 3kg toy have?
9514 1404 393
Answer:
216 J
Step-by-step explanation:
The kinetic energy is given by ...
KE = 1/2mv^2
KE = (1/2)(3 kg)(12 m/s)^2 = 216 kg·m²/s² = 216 J
The kinetic energy of the toy is 216 joules.
__
Its potential energy depends on its height.
Answer:216 J Is the answer I just checked
Step-by-step explanation:
In the United States, 13 out of 20 cans are recycled. What percent of cans are recycled?
Answer:
65%
Step-by-step explanation:
Divide 13 by 20:
0.65
Times 100:
65
So 65%
The lifetime of a battery has the exponential distribution with mean 6.9 months. If it fails, I immediately replace it with another – if I have one. I need a year’s supply of the battery. In addition to the one in the flashlight, how many extra should I bring with me? 1, 2, or 3?
Answer:
You should carry 2 other batteries.
Step-by-step explanation:
If the average life of each battery tends to last 6.9 months, and what you need is a year of battery supply, you probably need to carry two batteries.
The battery inside the flashlight has probably already spent much of its half-life.
We are told that the flashlight already has a battery inside, the batteries are discharged inside the flashlight even if the flashlight is not being used.
Which means that the average life of that battery is wearing out.
It would take 2 more batteries in addition to the one already inside the flashlight to supply the need for these for a year.
Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about how many candles can she make after melting the block of wax? Round to the nearest tenth.
Find the volume of the candle using the formula for volume of a cylinder:
V = PI x r^2 x h = 3.14 x 3.4^2 x 6 = 217.79 cubic cm. Round to 217.8
Find the volume of the block: V = l x w x h = 15 x 12 x 18 = 3,240 cubic cm.
Now divide the volume of the block by the volume of a candle:
3240 / 217.8 = 14.88
Round to the nearest tenth = 14.9 candles.
Number of cylindrical candles after melting the rectangular block is 14.9.
What is cylinder?" Cylinder is a three dimensional surface formed by two parallel lines and two parallel circular base."
What is volume?" Volume is the total amount of space occupied by any 3 dimensional body."
Formula used
Volume of a cylinder = π r²h
Volume of a rectangular block = length × width ×height
According to the question,
Height of a cylinder = 6cm
radius of cylinder = 3.4cm
Length of a block = 15cm
width of a block = 12cm
height of a block = 18cm
Volume of a cylinder = [tex]3.14[/tex]× [tex](3.4)^{2}[/tex]×[tex]6[/tex]
=217.8 cubic cm
Volume of rectangular block = [tex]15[/tex] × [tex]12[/tex] × [tex]18[/tex]
= 3240 cubic cm
Number of candles = Volume of rectangular block / volume of cylinder
= [tex]\frac{3240}{217.8}[/tex]
= 14.88
= 14.9 (nearest tenth)
Hence the Number of cylindrical candles after melting the rectangular block is 14.9.
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can anybody help please?
Answer:
AB = 24 cm, BC = 6 cm
Step-by-step explanation:
The perimeter of PQRS = 2(8) + 2(2) = 16 + 4 = 20 cm
Given the perimeter of ABCD is 60 cm, then
scale factor = [tex]\frac{60}{20}[/tex] = 3
That is the dimensions of ABCD are 3 times those of PQRS, so
AB = 3 × 8 cm = 24 cm
BC = 3 × 2 cm = 6 cm
Answer:
AB = 24 cm.
BC = 6 cm.
Step-by-step explanation:
The outer rectangle is a dilation of the inner rectangle centered at the origin. Therefore:
The inner and outer rectangles are similar so the ratio of length to width of the outer rectangle is the same as the ratio of length to width of the inner one.
so AB / BC = 8/2 = 4
AB = 4BC -------------(1)
Also as the perimeter = 60
2AB + 2BC = 60.............(2)
Substituting AB = 4BC in equation (2):
2(4BC) + 2BC = 60
10BC = 60
BC = 6 cm.
and AB = 4*6 = 24 cm.
Identify the equation that represents the line of best fit on this scatter plot.
Answery
y= 2x+2
Step-by-step explanation:
I need help with this question about similar triangles
Answer:
ST = 12 units
Step-by-step explanation:
Δ RAS and Δ TES are similar triangles, thus the ratios of corresponding sides are equal, that is, noting that SR = 21 - ST, thus
[tex]\frac{ST}{SR}[/tex] = [tex]\frac{TE}{AR}[/tex] substitute values
[tex]\frac{ST}{21-ST}[/tex] = [tex]\frac{8}{6}[/tex] ( cross- multiply )
6ST = 8(21 - ST)
6ST = 168 - 8ST ( add 8ST to both sides )
14ST = 168 ( divide both sides by 14 )
ST = 12
A dealer sold a car to a man and made
a propit of 15%. The man then sold it
to a woman for 120,175 at a loss of 5%. How much did the dealer buy the car
Answer:
110.000
Step-by-step explanation:
Selling at a loss of 5% means it was sold for 95% of its purchase value.
If 120.175 equals to 95% then 100%(the purchase value) is 126.500.
120.175/95*100 = 126.500 (plenty of other ways to calculate this, but this is one of the easiest)
So now we change to the dealer's perspective, he sold the car for 126.500 and for him that meant a 15% profit. If the purchase value is 100%, then the sale value is 115% (purchase value+15% profit)
Calculate the purchase value:
126.500/115*100 = 110.000
The dealer bought the car for 110.000, sold it for 126.500, so he made a profit of 16.500 (126.500-110.000) which is a 15% profit (15% of 110.000 is 16.500).
Complete the steps to derive the standard form of the equation of a
circle centered at Alh, k).
1. What is segment AB to circle A?
Check
Answer:
a radius
x-h
y-k
x-h, y-k
Step-by-step explanation:
The standard form of the circle is given by ( x - h )² + ( y - k )² = r²
What is a Circle?A circle is a closed figure in which the set of all the points in the plane is equidistant from a given point called “center”. Every line that passes through the circle forms the line of reflection symmetry. Also, the circle has rotational symmetry around the center for every angle
The circumference of circle = 2πr
The area of the circle = πr²
where r is the radius of the circle
The standard form of a circle is
( x - h )² + ( y - k )² = r²,
where r is the radius of the circle and (h,k) is the center of the circle.
The equation of circle is ( x - h )² + ( y - k )² = r²
For a unit circle , the radius r = 1
x² + y² = r² be equation (1)
Now , for a unit circle , the terminal side of angle θ is ( cos θ , sin θ )
Given data ,
Let the equation of circle be C
Let the center of the circle be represented as A ( h , k )
Now , the radius of the circle is = r units
And , standard form of a circle is
( x - h )² + ( y - k )² = r²
when it is a unit circle , the center is at ( 0 , 0 )
S0 , the equation of circle is x² + y² = r²
Hence , the standard form of circle is ( x - h )² + ( y - k )² = r²
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How do I solve this
Answer:
c = 30 inch
Step-by-step explanation:
perpendicular (p) = 18 inch
base(b) = 24 inch
hypotenuse (h) = ?
we know by using Pythagoras theorem
h² = p² + b²
h² = 18² + 24²
h² = 324 + 576
h² = 900
h= √900
therefore h = 30 inch
Hope it will help you. :)
If you flip a coin 300 times how many times would you expect the coin to lands head up ? Explain your reasoning (write in complete, mathematically precise sentences).
Answer: you would expect it to flip about 150 times heads.
Step-by-step explanation: A coin has two sides. There's an equal chance for each side. So 50-50 chance.
300 divided by 2 is 150
(Sorry if it's not the best mathematical way to say it)
Last fall, a middle school took a group of students on a trip to Space Camp, which is 360 miles away. The bus that they took to Space Camp traveled at a constant speed, and it took them 6 hours to get from the school to the camp. How many miles did the bus travel in 1 hour?
Answer:
The bus travelled 60 miles in 1 hour.
Step-by-step explanation:
To solve this problem we need to calculate the average speed at which the bus travelled. This is given by the distance travelled divided by the time that they took to get there. So we have:
average speed = distance/time
average speed = 360/6
average speed = 60 miles/h
Therefore in 1 hour they travelled 60 miles.
a loaded truck is traveling 20 mph faster than a freight train. In the time it takes the train to travel 160 miles, the truck travels 240 miles. find the speed of the truck
To find the speed of the truck, start by assigning the speed of the train as 'x', then the truck's speed is 'x + 20'. Set up an equation based on the principle of time = distance/speed, make them equal to each other then solve for 'x'. The truck's speed will be 'x + 20'.
Explanation:Let's call the speed of the freight train 'x' (in mph). We know from the question, that the loaded truck is traveling 20 mph faster than the train, so the truck's speed is 'x + 20' mph.
Given that the time it takes for the train to travel 160 miles is the same time it takes the truck to travel 240 miles, we can write an equation based on the principle that time equals distance divided by speed.
For the train, time = 160/x and for the truck, time = 240/(x+20). As both times are equal, we can write the equation as:
160/x = 240/(x + 20).
Solving this equation for 'x' will give us the speed of the freight train in mph. Then, by adding 20 to 'x', we can find the speed of the truck.
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The temperature inside a freezer was –18 °C when the freezer was accidentally unplugged. After that, the temperature increased by 2 °C per hour. This situation can be modeled by an equation of the form y = mx + b , where x = the time in hours that the freezer was been unplugged and y = the temperature in degrees Celsius. What is the value of b in this equation?
Answer:
-18
Step-by-step explanation:
b is where the line crosses the y-axis, or where x=0. Since it starts at -18, then that would be the b.
A right triangle has a side with length 20 meters, hypotenuse with length 25 meters, and side labeled b. StartRoot 5 EndRoot m 5 m StartRoot 45 EndRoot m 15 m
Answer:
15 m
Step-by-step explanation:
25² = 20² + b²
625 = 400 + b²
b² = 225
b = 15
it is 15 hope this helped and good luck...
Anyone can help me with all of these answers 1-10 ( 18 Points ) ?
Answer:
1) 14
2) 8
3) 5
4) 11
5) 10
6) 6
7) 10
8) 17
9) 26
10) 18
Step-by-step explanation:
Merry Early Christmas :)
5. Alisha's average math test score was 82. Which of the following students has the same average
math test score as Alisha?
F. Jenny earned 492 points on 6 tests.
G. Frankie earned 352 points on 4 tests.
H. Benicio earned 468 points on 6 tests.
1. Dontonio earned 344 points on 4 tests.
(20 Points)
Answer:
F. Jenny earned 492 points on 6 tests.
Jenny has the same average
math test score as Alisha i.e. 82.
Step-by-step explanation:
Points earned by Jenny = 492
no. of maths tests attempted by Jenny = 6
average maths test score = 492/6
=82
IF I DOG WALK AND FOR EACH DOG I MAKE $15 HOW MANEY DOGS DO I HAVE TO WALK TO GET $100
Answer:
7
Step-by-step explanation:
what’s the coordinate grid ?
Answer:
linear and lateral
Step-by-step explanation:
thsts it
Final answer:
A coordinate grid, also known as a Cartesian coordinate system, is a mechanism used in mathematics to define every point in a plane with a pair of numerical values indicating horizontal and vertical positions from an origin. Systems like UTM and SPC are derived from this concept, using a grid to simplify plotting and calculations on a plane surface.
Explanation:
A coordinate grid, or a Cartesian coordinate system, is a tool used in mathematics to specify each point uniquely in a plane by a pair of numerical coordinates. These are typically written as an ordered pair (x, y) where 'x' represents the horizontal displacement and 'y' represents the vertical displacement, both measured from a specific origin point. This system of geometry was developed by René Descartes, hence the name Cartesian.
For example, the origin of a coordinate system is the starting point where both x and y are zero, designated as (0,0). If we move to the right, x increases, and if we move up, y increases. Conversely, moving to the left decreases x and moving down decreases y. So, for instance, the point (3,4) is located 3 units to the right of the origin and 4 units above it.
Plane coordinate systems, like the Universal Transverse Mercator (UTM) and the State Plane Coordinate (SPC) systems, simplify calculations by using a grid pattern superimposed on a flattened representation of the Earth's surface, facilitating easier and more precise location plotting than spherical coordinate systems like latitude and longitude.
Why a quadratic function f whose zeros are 2 and 7
A quadratic function whose zeros are 2 and 7 is [tex]f(x) = x^2 - 9x+14[/tex]
If the zeros of a quadratic function are a and b, the quadratic function will be given as:
f(x) = (x - a)(x - b)
For the function f(x) whose zeros are 2 and 7:
f(x) = (x - 2)(x - 7)
Expand the equation
[tex]f(x)=x^2-7x-2x+14\\\\f(x) = x^2 - 9x+14[/tex]
Therefore, a quadratic function whose zeros are 2 and 7 is [tex]f(x) = x^2 - 9x+14[/tex]
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Final answer:
A quadratic function with zeros at 2 and 7 can be constructed by multiplying (x - 2) and (x - 7), leading to the function f(x) = (x - 2)(x - 7). A constant 'a' scales the function, which, if not specified, can be assumed to be 1 for simplicity.
Explanation:
The question asks about constructing a quadratic function f with zeros at 2 and 7. In mathematics, a quadratic function is of the form f(x) = ax2 + bx + c, where a, b, and c are constants and a ≠ 0.
To construct a quadratic function with zeros 2 and 7, we utilize the fact that if 2 and 7 are zeros, the function must be divisible by (x - 2) and (x - 7).
Hence, f(x) can be written as f(x) = a(x - 2)(x - 7), where a is a non-zero constant that scales the function.
To determine a, more information about the function is needed, such as a specific value f(x) takes at a certain point.
In the absence of additional information, we can assume a = 1 for simplicity, which gives us the specific quadratic function f(x) = (x - 2)(x - 7).
This function will cross the x-axis at x = 2 and x = 7, making them its zeros.
The stopping distance of a car is modeled by the function d = 0.05r(r + 2) where d is the stopping distance of the car measured in feet and r is the speed of the car in miles per hour. If skid marks left on the road are 75 feet long, how fast was the car traveling?
Answer:
The car was moving approximately at a speed of 37.74 miles per hour.
Step-by-step explanation:
We are given the following in the question:
The stopping distance of a car is given by
[tex]d = 0.05r(r + 2)[/tex]
where d is the stopping distance in feet and r is speed of the car in miles per hour.
The stopping distance is 75 feet, we have to find the speed of the car,
Putting d = 75 in the equation, we get,
[tex]75 = 0.05r(r + 2)\\r^2 + 2r = 1500\\r^2 + 2r-1500 = 0\\\\r =\dfrac{-2\pm \sqrt{4-(4)(1)(-1500)}}{2}\\\\r\approx -39.74,37.74[/tex]
Thus, the car was moving approximately at a speed of 37.74 miles per hour.
GEOMETRY! Show your work, please.
Answer:
90°
Step-by-step explanation:
By inscribed angle theorem:
[tex]m\angle STU = \frac{1}{2} \times 180 \degree = 90 \degree[/tex]
Find the midpoint of the line segment with end coordinates of:
(
0
,
2
)
and
(
2
,
8
)
The midpoint is just the average of the coordinates,
M = (1/2) ( (0,2) + (2,8) ) = ((0+2)/2, (2+8)/2) = (1, 5)
Answer: (1,5)
Answer:
(1,5)
Step-by-step explanation:
(0,2) and (2,8)
Midpoint: (0+2)/2, (2+8)/2
2/2, 10/2
(1,5)
Please help me answer this
Answer:
B. 3
Step-by-step explanation:
Students set a goal of collecting 900 cans for the canned food drive. The number of cans they have collected so far is 82% of their goal. How many cans have they collected so far?
Answer:
738 cans.
Step-by-step explanation:
To calculate 82% of 900, take 0.82 *900. Your answer will be 738.
Answer:
738 cans
Step-by-step explanation:
900 x .82=738
in the adjoining figure P A and PB are tangents from P to a circle with Centre O if angle APB is equal to 40 degree then find angle ACB
Step-by-step explanation:
40° + 90° + 90° + angle ACB = 360°
220° + angle ACB = 360°
angle ACB = 360° - 220°
Therefore angle ACB = 140°
Hope it will help :)
Final answer:
To find angle ACB, determine angle AOB by subtracting the tangents' right angles and angle APB from 360 degrees, which results in 140 degrees. Then, using the Inscribed Angle Theorem, angle ACB is half the measure of angle AOB, yielding 70 degrees.
Explanation:
The student is asking how to find the angle ACB when given that PA and PB are tangents to a circle with centre O, and angle APB is 40 degrees. To find angle ACB, which is the angle at the circle subtended by the chord AB, we first determine the measure of angle AOB, which is central and subtended by AB as well. Since the tangents from a point outside a circle form a right angle with the radius at the point of contact, we have two right triangles OAP and OBP, each with an angle of 90 degrees. These right angles add to the given angle APB to form a full angle around point P which is 360 degrees. Knowing angle APB is 40 degrees, we can find angle AOB by subtracting the sum of the two right angles and angle APB from 360 degrees.
Angle AOB = 360 degrees - (90 degrees + 90 degrees + 40 degrees) = 360 degrees - 220 degrees = 140 degrees
Finally, since angle ACB is inscribed and angle AOB is central, both subtended by the same chord AB, angle ACB is half the measure of angle AOB by the Inscribed Angle Theorem. Therefore, angle ACB is 70 degrees.
Angle ACB = ½ * Angle AOB = ½ * 140 degrees = 70 degrees
Step 1: 5 = negative 2 (negative StartFraction 3 Over 10 EndFraction) + b. Step 2: 5 = StartFraction 6 Over 10 EndFraction + b. Step 3: 5 minus StartFraction 6 Over 10 EndFraction = StartFraction 6 Over 10 EndFraction minus StartFraction 6 Over 10 EndFraction + b. Step 4: StartFraction 50 Over 10 EndFraction minus StartFraction 6 Over 10 EndFraction = b. Step 5: StartFraction 44 Over 10 EndFraction = b. Step 6: y = negative 2 x minus 4.4.
A student used the slope-intercept form to write the equation of a line that has a slope of -2 and passes through the point (StartFraction negative 3 Over 10 EndFraction, 5). The student’s work is shown to the left.
Analyze the steps to identify the error.
Step1: student switched the x and y values
Step 3: student needed to add StartFraction 6 Over 10 EndFraction to both sides
Step 5: student subtracted incorrectly
Step 6: student needed to add the b value
Answer:
D.) Step 6: student needed to add the b value.
Step-by-step explanation:
I got it wrong and it gave me the right answer, which is what i provided, D.
Answer: Step6
Step-by-step explanation:
how do i covert a fraction into a decimal.
Answer:
By dividing the top number by the bottom.
Step-by-step explanation:
ummmmm
To convert a fraction into a decimal, divide the fraction's numerator by its denominator. Multiplying a decimal by 100 converts it to a percent, and dividing a percent by 100 converts it back into a decimal. This process is used to manipulate numerical values for various mathematical applications.
Explanation:Converting a Fraction to a Decimal
To convert a fraction into a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert 3/4 to a decimal, you divide 3 by 4, which equals 0.75. If the fraction is not in simplest form, you may first simplify it before dividing. This is a basic mathematical skill that you will use in many areas of math.
To convert a decimal to a percent, you multiply the decimal by 100. For instance, 0.75 multiplied by 100 equals 75%, because you are essentially finding how many 'parts out of 100' are present in the decimal.
In the case of converting percents, to turn a percent into a decimal, you divide by 100. So, taking 75% and dividing by 100 would give you 0.75, which is the original decimal we started with.
Another example is converting 0.0479 into a decimal that's a number between 1 and 10. To do this, we would move the decimal point to the right two places to get 4.79, as specified by the conversion factor which helps in adjusting units or scaling numbers without changing the value.