Final answer:
To find the intersection points of the paths of two particles, one needs to set their position functions equal and solve for the variable t. If no solution exists, then the intersection points do not exist, and the answer is DNE.
Explanation:
To find the points where the paths of two particles intersect, we set their respective position functions r1(t) and r2(t) equal to each other and solve for t. The position functions given are:
r1(t) = (t, t^2, t^3)
r2(t) = (1 + 4t, 1 + 16t, 1 + 52t).
Their paths intersect when:
x1 = x2
y1 = y2
z1 = z2.
By solving these equations, we find the common t values that make both position vectors equal. If there is no common t that satisfies all three conditions, then the particles never collide, and the answer is DNE (Does Not Exist).
Which of the following functions has the same horizontal asymptote and
range as the function graphed below?
A. f(x) = 2^x+2 + 3
B. f(x) = 2^x+2 + 2
C. f(x) = 2^x+2 - 2
D. f(x) = 2^x+2 - 3
Answer:
C. f(x) = 2^x+2 - 3
Step-by-step explanation:
Looking at the graph and answer choices, it's obvious that the only change is the vertical shift. Since we know that the smallest value of 2^x+2 is 0, we can infer that the vertical shift will be -3 to match the horizontal asymptote of y=-3.
A player of a video game is confronted with a series of four opponents and an 80% probability of defeating each opponent. Assume that the results from opponents are independent (and that when the player is defeated by an opponent the game ends).
A. What is the probability that a player defeats all four opponents in a game?
B. What is the probability that a player defeats at least two opponents in a game?
C. If the game is played three times, what is the probability that the player defeats all four opponents at least once?
Answer:
(a) 0.4096
(b) 0.64
(c) 0.7942
Step-by-step explanation:
The probability that the player wins is,
[tex]P(W)=0.80[/tex]
Then the probability that the player losses is,
[tex]P(L)=1-P(W)=1-0.80=0.20[/tex]
The player is playing the video game with 4 different opponents.
It is provided that when the player is defeated by an opponent the game ends.
All the possible ways the player can win is: {L, WL, WWL, WWWL and WWWW)
(a)
The results from all the 4 opponents are independent, i.e. the result of a game played with one opponent is unaffected by the result of the game played with another opponent.
The probability that the player defeats all four opponents in a game is,
P (Player defeats all 4 opponents) = [tex]P(W)\times P(W)\times P(W)\times P(W)=[P(W)]^{4} =(0.80)^{4}=0.4096[/tex]
Thus, the probability that the player defeats all four opponents in a game is 0.4096.
(b)
The probability that the player defeats at least two opponents in a game is,
P (Player defeats at least 2) = 1 - P (Player losses the 1st game) - P (Player losses the 2nd game) = [tex]1-P(L)-P(WL)[/tex]
[tex]=1-(0.20)-(0.80\times0.20)\\=1-0.20-0.16\\=0.64[/tex]
Thus, the probability that the player defeats at least two opponents in a game is 0.64.
(c)
Let X = number of times the player defeats all 4 opponents.
The probability that the player defeats all four opponents in a game is,
P(WWWW) = 0.4096.
Then the random variable [tex]X\sim Bin(n=3, p=0.4096)[/tex]
The probability distribution of binomial is:
[tex]P(X=x)={n\choose x}p^{x} (1-p)^{n-x}[/tex]
The probability that the player defeats all the 4 opponents at least once is,
P (X ≥ 1) = 1 - P (X < 1)
= 1 - P (X = 0)
[tex]=1-[{3\choose 0}(0.4096)^{0} (1-0.4096)^{3-0}]\\=1-[1\times1\times (0.5904)^{3}\\=1-0.2058\\=0.7942[/tex]
Thus, the probability that the player defeats all the 4 opponents at least once is 0.7942.
Final answer:
The probability of defeating all four opponents in one game is 40.96%. The probability of defeating at least two opponents is 87.04%. Playing the game three times, the probability of defeating all four opponents at least once is 88.47%.
Explanation:
To calculate the probability of different outcomes when playing a video game against four opponents, we'll use basic probability rules and assumptions given in the question.
A. Probability of Defeating All Four Opponents
The probability of defeating each opponent is 80% or 0.8. Since the fights are independent, to find the probability of defeating all four, we multiply the probabilities together:
0.8 × 0.8 × 0.8 × 0.8 = 0.4096 or 40.96% chance of defeating all four opponents.
B. Probability of Defeating At Least Two Opponents
To find the probability of defeating at least two opponents, we must consider all possible combinations of winning 2, 3, or 4 opponents' fights, and add those probabilities together. Let W represent a win and L represent a loss:
P(WWLL) + P(WLWL) + P(WLLW) + P(LWWL) + P(LWLW) + P(LLWW)We already calculated P(WWWW) as 0.4096. The other probabilities can be calculated using similar multiplication of respective individual probabilities (0.8 for W, 0.2 for L). After calculation, we sum these to find the cumulative probability, which is 0.8704 or 87.04%.
C. Probability of Defeating All Four Opponents At Least Once Over Three Games:
The probability of not defeating all four opponents in one game is 1 - P(WWWW) = 0.5904. The events in each game are independent, so the probability of not defeating all four opponents in all three games is (0.5904)^3. Therefore, by subtracting this result from 1, we get the probability of defeating all opponents at least once: 1 - (0.5904 × 0.5904 × 0.5904) = 0.8847 or 88.47%.
What’s the value of x?
Answer:
98 degrees
Step-by-step explanation:
m<H = 45, m<F = 53
180 = 45 + 53 + (180-x)
82 = 180-x
x+82 = 180
x = 98
Answer:
Step-by-step explanation:
The sum of the angles in a triangle is 180 degrees. This means that
Angle F + angle G + angle H = 180
Therefore,
53 + 45 + angle G = 180
98 + angle G = 180
Subtracting 98 from the left hand side and the right hand side of the equation, it becomes
98 - 98 + angle G = 180 - 98
angle G = 82 degrees
The sum of angles on a straight line is 180 degrees. Therefore,
x + 82 = 180
Subtracting 82 from the left hand side and the right hand side of the equation, it becomes
x + 82 - 82 = 180 - 82
x = 98 degrees
Issa jogged two-thirds of the way home from school. Then he was tired, so he walked the remaining 3{,}200\text{ m}3,200 m3, comma, 200, start text, space, m, end text. How many kilometers did Issa travel from school to his house? kilometers
Answer:
9.6
Step-by-step explanation:
The question is :
Solution;
Let total distance from school to home = 'x' meters
Issa jogged 2/3rd of 'x'= [tex]\frac{2x}{3}[/tex]
So total distance from school to home = [tex]\frac{2x}{3} +3200[/tex]
But total distance from school to home = [tex]x[/tex]
[tex]==> \frac{2x}{3} + 3200 = x[/tex]
[tex]==> 3200= \frac{x}{3}[/tex]
[tex]==> 9600=x[/tex]
So total distance from school to home is 9600 meters
We convert it into km now
we know 1000 meters = 1km
==> 9600 meters = 1/100 x 9600= 9.6 km
Answer:
9.6 or (48\5
Step-by-step explanation:
Dan's business loses $9 each day. Which equation would be used to show how much money Dan's business has lost in a week? A. -9 × 7 B.-9 × (-7) C.9 × 7 D.9 × (-7)
Answer:
[tex]C. \: 9 \times 7[/tex]
the parallelogram shown below has an area of 54 units. find the missing height
Answer:
h = 6 units.
Step-by-step explanation:
Given:
Area of the parallelogram = [tex]54\ units^2[/tex].
Base = 9 units.
We need to find the missing height.
Solution:
We know that the area of the parallelogram is represented by below formula.
[tex]A = bh[/tex] -----------(1)
Where:
A = Area of the parallelogram.
h = height of the parallelogram.
b = Base
Since, base and area is known. So, we substitute these values in equation 1.
[tex]54 = 9\times h[/tex]
[tex]h = \frac{54}{9}[/tex]
h = 6 units.
Therefore, height of the parallelogram h = 6 units.
The perimeters of two 30-60-90 triangles are in the ratio 1:2. If the length of the hypotenuse of the larger triangle is 20 cm, find the length of the longer leg of the smaller triangle.
Answer:
The answer to your question is 5[tex]\sqrt{3}[/tex]
Step-by-step explanation:
Data
Proportion 1:2
Hypotenuse of the larger triangle = 20
length of the longer leg of the smaller triangle = ?
Process
1.- Remember the proportions of a 30- 60 - 90 triangle
hypotenuse = 2x
short leg = x
long leg = x[tex]\sqrt{3}[/tex]
2.- Use the previous information to find the lengths of the larger triangle
hypotenuse = 20 = 2x
short leg = x = 10
large leg = 10[tex]\sqrt{3}[/tex]
3.- Use the previous information to find the lengths of the smaller triangle
Proportion 1:2
hypotenuse = 10
short leg = x = 5
long leg = 5[tex]\sqrt{3}[/tex]
Answer:
It’s 5 radical 3
Step-by-step explanation:
PLEASE HELP ASAP!!! I NEED CORRECT ANSWERS ONLY PLEASE!!!
Find m∠R.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠R = °
Answer:
Therefore,
[tex]m\angle R = 69.4\°[/tex]
Step-by-step explanation:
Given:
In Right Angle Triangle PQR at ∠Q = 90° such that
RQ = 3 ....Adjacent side of angle R
PQ = 8 ....Opposite side of angle R
To Find:
m∠R = ?
Solution:
In Right Angle Triangle PQR, Tan Identity,
[tex]\tan R= \dfrac{\textrm{side opposite to angle R}}{\textrm{side adjacent to angle R}}[/tex]
Substituting the values we get
[tex]\tan R= \dfrac{PQ}{QR}=\dfrac{8}{3}=2.6666\\\\\angle R=\tan^{-1}(2.666)=69.439=69.4\°[/tex]
Therefore,
[tex]m\angle R = 69.4\°[/tex]
Hi does anyone know how to solve this question if so please show the working out.
Explanation:
The Law of Sines is your friend, as is the Pythagorean theorem.
Label the unmarked slanted segments "a" and "b" with "b" being the hypotenuse of the right triangle, and "a" being the common segment between the 45° and 60° angles.
Then we have from the Pythagorean theorem ...
b² = 4² +(2√2)² = 24
b = √24
From the Law of Sines, we know that ...
b/sin(60°) = a/sin(θ)
y/sin(45°) = a/sin(φ)
Solving the first of these equations for "a" and the second for "y", we get ...
a = b·sin(θ)/sin(60°)
and ...
y = a·sin(45°)/sin(φ)
Substituting for "a" into the second equation, we get ...
y = b·sin(θ)/sin(60°)·sin(45°)/sin(φ) = (b·sin(45°)/sin(60°))·sin(θ)/sin(φ)
So, we need to find the value of the coefficient ...
b·sin(45°)/sin(60°) = (√24·(√2)/2)/((√3)/2)
= √(24·2/3) = √16 = 4
and that completes the development:
y = 4·sin(θ)/sin(φ)
A baby otter was born 3/4 of a month early at first it's weight was 7/8 kg which is 9/10 kg less than the average weight of newborn otter in the aquarium what is the average weight of a newborn otter
Answer:
The average weight of the newborn otter is [tex]\frac{71}{40}\ kg.[/tex]
Step-by-step explanation:
Given:
A baby otter was born 3/4 of a month early at first it's weight was 7/8 kg which is 9/10 kg less than the average weight of newborn otter in the aquarium.
Now, to find the average weight of the newborn otter.
Let the average weight of the newborn otter be [tex]x.[/tex]
It's weight was = [tex]\frac{7}{8} \ kg.[/tex]
It's weight is less than the average weight of newborn by = [tex]\frac{9}{10} \ kg.[/tex]
According to question:
[tex]x-\frac{9}{10}=\frac{7}{8}[/tex]
Adding both sides by [tex]\frac{9}{10}[/tex] we get:
[tex]x=\frac{9}{10} +\frac{7}{8}[/tex]
[tex]x=\frac{36+35}{40}[/tex]
[tex]x=\frac{71}{40} \ kg.[/tex]
Therefore, the average weight of the newborn otter is [tex]\frac{71}{40}\ kg.[/tex]
NASA is sending a probe to Alpha Centauri and then to Sirius. A problem with the probe is noticed while it is at Alpha Centauri, so it must go back to Erth before going to Sirius. Alpha Centauri is 4.3 light-years away from Earth and Sirius is 8.6 light-years away. The probe is traveling at 18.03 km/s, there are 1.58125 x 10-5 light-years in one astronomical unit. How long will the probe have been traveling from when it first leaves Earth to when it arrives at Sirius?
Answer:
Time = 9.0252 *10^12 s
Step-by-step explanation:
Given:
- The distance from Earth to Alpha Centauri = 4.3 light years
- The distance from Earth to Sirius = 8.6 light years
- Speed of the probe is V = 18.03 km/s
- 1 AU = 1.58125 x 10-5 light-years
Find:
How long will the probe have been traveling from when it first leaves Earth to when it arrives at Sirius?
Solution:
- We will track probe for each destination it reaches one by one:
Earth ------> Alpha Centauri d_1 = 4.3 light years
Alpha Centauri ------> Earth d_2 =4.3 light years
Earth ------> Sirius d_3 = 8.6 light years
Total distance D = 17.2 light years.
- Now we know the total distance traveled by the probe is D. We will convert the distance into km SI units:
1 AU ------------------> 1.58125 x 10-5 light-years
x AU ------------------> 17.2 light years.
- Using direct proportions
x = 17.2 / (1.58125 x 10-5) = 1087747.036 AU
Also,
1 AU ---------------------> 149597870700 m
1087747.036 AU ----> D m
- Using direct proportions
D = 1087747.036*149597870700 = 1.62725*10^17 m
- Now use the speed - distance - time formula to compute the total time taken:
Time = Distance traveled (D) / V
Time = 1.62725*10^17 / 18.03*10^3
Answer: Time = 9.0252 *10^12 s
What is the value of n?
Enter your answer in the box.
n =
m
Circle with two intersecting chords forming an x shape in the circle. The top left side of the x shape is labeled 5 meters. The top right side of the x shape is labeled 2 meters. The bottom left side of the x is labeled 15 meters.The bottom right side of the x is labeled n.
Answer:
6 meters
Step-by-step explanation:
Intersecting Chord Theorem: When two chords intersect each other inside a circle, the products of their segments are equal.
One chord is divided into two segments with lengths of 15 m and 2 m.
Anothe chord is divided into two segments with measures of 5 m and n m.
Therefore,
[tex]5\cdot n=15\cdot 2\\ \\5n=30\\ \\n=6\ m[/tex]
Answer:
6 m
Step-by-step explanation:
just took the test
The 1985 1985 explosion at a nuclear lab sent about 1000 kilograms of a radioactive element into the atmosphere. The function f left parenthesis x right parenthesis equals 1000 left parenthesis 0.5 right parenthesis Superscript StartFraction x Over 30 EndFraction f(x)=1000(0.5) x 30 describes the amount, f(x), in kilograms, of a radioactive element remaining in the area x years after 1985 1985. If even 100 kilograms of the radioactive element remains in the atmosphere, the area is considered unsafe for human habitation. Find f( 40 40) and determine if the area will be safe for human habitation by 2025 2025.
Answer:
Step-by-step explanation:
If I'm understanding this correctly, the rate of decay function is
[tex]f(x)=1000(.5)^{\frac{x}{30}}[/tex]
and we want to solve for the amount of element left after x = 40 years. That would make our equation
[tex]f(40)=1000(.5)^{1.33333333333}[/tex]
Multiply the repeating decimal by .5 to get
f(40) = 1000(.3968502631)
and f(40) = 396.85
So no, it's not safe for human habitation.
f(40) ≈ 793.7 kilograms. The area will not be safe for human habitation by 2025.
Explanation:For the decay, to find f(40), we substitute x=40 into the function f(x)=1000(0.5)x/30.
f(40)=1000(0.5)40/30
f(40)=1000(0.5)4/3
f(40) ≈ 1000(0.7937)
f(40) ≈ 793.7 kilograms
Since the question asks if the area will be safe for human habitation by 2025, we need to check if f(40) is less than or equal to 100 kilograms.
But the value of f(40) is about 793.7 kilograms, which is greater than 100 kilograms.
Therefore, the area will not be safe for human habitation by 2025.
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The average weight of 5 items is 24 pounds. If the total weight of 3 blue items is 39 pounds, what is the average weight of the remaining 2 non-blue items?
Answer: the average weight of the remaining 2 non-blue items is 81 pounds
Step-by-step explanation:
The formula for determining average is expressed as
Average = sum of each item/ total number of items
Let x represent the average weight of the remaining 2 non-blue items.
The average weight of 5 items is 24 pounds. If the total weight of 3 blue items is 39 pounds, it means that
(x + 39)/5 = 24
x + 39 = 5 × 24 = 120
x = 120 - 39
x = 81
Looking into the sky one night, Tori wondered how far into outer space she would get if she drove a car for 3.21 × 103 hours at a rate of 70mph. She calculated and determined that a car travelling at 70 mph covers approximately 1.13 × 105 meters per hour. Tori wrote this expression to determine the distance she would travel into outer space. (3.21 × 103)(1.13 × 105) Estimate the distance travelled
Answer:
[tex]\large\boxed{\large\boxed{3\times 10^8meters}}[/tex]
Explanation:
One of the applications of scientific notation is to make estimations rounding the whole part of the numbers, i.e. the digits before the decimal point, to the nearest integer, and adding the exponents of the powers of base 10.
Here, you must estimate this product of two numbers written is scientific notation:
[tex](3.21\times 10^3)(1.13\times 10^5)[/tex]
Then, for an estimation you round 3.21 to 3 and 1.13 to 1, then multiply 3 × 1 = 3. That will be the coefficient of your new power of 10.
The power or exponent will be the sum of the powers of the numbers that are being multiplied, i.e. 3 + 5 = 8.
And the result is [tex]3\times 10^8[/tex]
The unit is meters, so you write your answer as: [tex]3\times 10^8meters[/tex]
Answer:
b. 3*10^8
Step-by-step explanation:
Compare the ratios of sauce to dough. Which of the
following are true? Check all that apply.
Adrian's recipe has a greater ratio of sauce to
dough.
Juliet's recipe has a greater ratio of sauce to dough.
Adrian's and Juliet's recipes have equal ratios of
sauce to dough.
The ratios can be compared by comparing 15 to 18
because 30 is in both dough columns.
Answer:b also d
Step-by-step explanation:
Adrian's recipe has a greater ratio of sauce to dough, and the ratios can be compared by comparing 15 to 18 because 30 is in both dough columns.
In Adrian's recipe, the ratio of sauce to dough is 15:30, which simplifies to 1:2.
In Juliet's recipe, the ratio of sauce to dough is 18:30, which also simplifies to 1:2.
Since both ratios simplify to the same value of 1:2, it means that Adrian's and Juliet's recipes have equal ratios of sauce to dough.
Therefore, the statement "Adrian's and Juliet's recipes have equal ratios of sauce to dough" is true.
The reason we can compare the ratios by looking at 15 and 18 is because both of these values represent the amount of sauce, and 30 represents the amount of dough in both recipes.
By comparing the sauce portions directly, we can determine that the ratios are indeed equal.
It's important to note that the absolute values of the sauce and dough quantities are less relevant in this context than their proportions, which are expressed in the ratios.
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Miguel orders 595 candy bars. They come in 7 boxes. How many candy bars are in each box? How many candy bars will he have left if he gives 3 boxes to his friend?
There are 85 candy bars in each box, and he will have 340 candy bars left after he gives three boxes to a friend.
Answer: he has 340 candy bars left.
Step-by-step explanation:
The total number of candy bars that
Miguel ordered is 595.
They come in 7 boxes. Assuming each box contains equal number of candy bars. This means that the number of candy bars in each box would be
595/7 = 85 candy bars
If he gives 3 boxes of candy bars to his friend, it means that the number of candy bars that he gave to his friend is
85 × 3 = 255 candy bars
Therefore, the number of candy bars that he has left is
595 - 255 = 340
A company wants to determine the amount of a vitamin mix that can be enclosed in a capsule like the one shown. The capsule has a radius of 3millimeters (mm) and a length of 10 mm. How much vitamin mix is needed?
Answer:
226.08 mm³
Step-by-step explanation:
The figure below shows a capsule mad of;
Two hemisphere and an open cylinderTo determine the amount of vitamin that can be enclosed in the capsule, we determine the volume of the capsule;
First we determine the volume of the two hemisphere;Volume of a hemisphere = 2/3 πr³
Therefore; Taking π to be 3.14
Then;
Volume of the hemisphere = 2/3 × 3.14 × 3³
= 56.52 mm³
But, since there are two equal hemispheres;
Then;
Volume of hemispheres = 56.52 mm³ × 2
= 113.04 mm³
Second we determine the volume of the cylinderVolume of the cylinder is given by;
Volume = πr²h
Height = (10 mm - (2× 3mm)
= 4 mm
Thus, taking π to be 3.14
Then;
Volume of the cylinder = 3.14 × 3² × 4
= 113.04 mm³
Thus, Volume of the capsule = 113.04 mm³ + 113.04 mm³
= 226.08 mm³
I WILL GIVE BRAINLIEST! 30 POINTS!
Koji is installing a rectangular window in an office building. The window is 823 feet wide and 534 feet high.
The formula for the area of a rectangle is A=bh.
What is the area of the window?
Enter your answer as a mixed number in simplest form in the box.
$$
ft2 ( ? ) ( ? )
-------
( ? )
Answer:
Area = 823 * 534 = 439,482 ft^2
Step-by-step explanation:
A box with a square base and an open top is being constructed out of A cm2 of material. If the volume of the box is to be maximized, what should the side length of the base be? What should the height of the box be? What is the maximal volume of the box? Your answers should be in terms of A.
FInd: Side length(cm), Height(cm), and Volume(cm)
Answer:
Side length = [tex]\sqrt{\frac{A}{3} }[/tex] cm , Height = [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex] cm , Volume = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex] cm³
Step-by-step explanation:
Assume
Side length of base = x
Height of box = y
total material required to construct box = A ( given in question)
So it can be written as
A = x² + 4xy
4xy = A - x²
[tex]y = \frac{A - x^{2} }{4x}[/tex]Volume of box = Area x height
V = x² ₓ y
V = x² ₓ ( [tex]\frac{A - x^{2} }{4x}[/tex] )
V = [tex]\frac{Ax - x^{3} }{4}[/tex]
To find max volume put V' = 0
So taking derivative equation becomes
[tex]\frac{A - 3 x^{2} }{4} = 0[/tex]
A = 3 [tex]x^{2}[/tex]
[tex]x^{2}[/tex] = [tex]\frac{A}{3}[/tex]
x = [tex]\sqrt{\frac{A}{3\\} }[/tex]
put value of x in equation 1
y = [tex]\frac{A - \frac{A}{3} }{4\sqrt{\frac{A}{3} } }[/tex]
y = [tex]\frac{2 \sqrt{\frac{A}{3} } }{4 \sqrt{\frac{A}{3} } }[/tex]
y = [tex]\frac{1}{2} \sqrt{\frac{A}{3} }[/tex]
So the volume will be
V = [tex]x^{2}[/tex] × y
Put values of x and y from equation 2 & 3
V = [tex]\frac{A}{3} (\frac{1}{2} \sqrt{\frac{A}{3} } )[/tex]
V = [tex]\frac{A\sqrt{A}}{6\sqrt{3} }[/tex]
The side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]
Let the dimensions of the box be l and h, where l represents the base length and h represents the height.
The volume is calculated as:
[tex]\mathbf{V = l^2h}[/tex]
The surface area is:
[tex]\mathbf{A= l^2 + 4lh}[/tex]
Make h the subject
[tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]
Substitute [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex] in [tex]\mathbf{V = l^2h}[/tex]
[tex]\mathbf{V = l^2 \times \frac{A- l^2}{4l}}[/tex]
[tex]\mathbf{V = l \times \frac{A- l^2}{4}}[/tex]
[tex]\mathbf{V = \frac{Al- l^3}{4}}[/tex]
Split
[tex]\mathbf{V = \frac{Al}{4}- \frac{l^3}{4}}[/tex]
Differentiate
[tex]\mathbf{V' = \frac{A}{4}- \frac{3l^2}{4}}[/tex]
Set to 0
[tex]\mathbf{\frac{A}{4}- \frac{3l^2}{4} = 0}[/tex]
Multiply through by 4
[tex]\mathbf{A- 3l^2 = 0}[/tex]
Add 3l^2 to both sides
[tex]\mathbf{3l^2 = A}[/tex]
Divide both sides by 3
[tex]\mathbf{l^2 = \frac A3}[/tex]
Take square roots
[tex]\mathbf{l =\sqrt{ \frac A3}}[/tex]
Recall that: [tex]\mathbf{h = \frac{A- l^2}{4l}}[/tex]
So, we have:
[tex]\mathbf{h = \frac{A - \frac{A}{3}}{4\sqrt{A/3}}}[/tex]
[tex]\mathbf{h = \frac{\frac{2A}{3}}{4\sqrt{A/3}}}[/tex]
Divide
[tex]\mathbf{h = \frac{2\sqrt{A/3}}{4}}[/tex]
[tex]\mathbf{h = \frac{\sqrt{A/3}}{2}}[/tex]
Rewrite as:
[tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex]
Recall that:
[tex]\mathbf{V = l^2h}[/tex]
So, we have:
[tex]\mathbf{V = \frac A3 \times \frac{1}{2}\sqrt{\frac A3}}[/tex]
[tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]
Hence, the side length is [tex]\mathbf{l =\sqrt{ \frac A3}}[/tex], the height is [tex]\mathbf{h = \frac{1}{2}\sqrt{\frac A3}}[/tex] and the maximal volume is [tex]\mathbf{V = \frac A6 \sqrt{\frac A3}}[/tex]
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Consider a coordinate system in which the positive x axis is directed upward vertically. What are the positions of a particle (a) 5.0 m directly above the origin and (b) 2.0 m below the origin?
Answer:
a) (5,0)
b) (2,0)
Step-by-step explanation:
(a) A particle that lies 5.0 m directly above the origin would have its x-coordinate be 5 and its y-coordinate be 0. So (5,0).
(b) A particle that lies 2.0 m directly below the origin would have its x-coordinate be 2 and its y-coordinate be 0. So (2,0).
In a coordinate system with the positive x-axis directed upward, a particle located 5.0 m directly above the origin has a position of +5.0 m, and a particle 2.0 m below the origin has a position of -2.0 m.
Explanation:In this coordinate system, the position of a particle is indicated by its vertical position relative to the origin. Therefore:
Directly above the origin: If a particle is 5.0 m directly above the origin, this is represented as a positive number in this coordinate system. Thus, its coordinate is +5.0 m.Below the origin: If a particle is 2.0 m below the origin, this is represented as a negative number in this coordinate system. Thus, its coordinate is -2.0 m.Learn more about Coordinate System here:https://brainly.com/question/32885643
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A ball is dropped from a height of 16 feet. The function f(x)=16(0.67)^x gives the height in feet of each bounce, where x is the bounce number. What will be the height of the second bounce rounded to the nearest tenth of a foot?
Answer:
Step-by-step explanation:
This is an exponential function:
[tex]y=16(\frac{2}{3})^x[/tex] that tells us that the initial height of the ball is 16 feet and that after each successive bounce the ball comes up to 2/3 its previous height. We are looking for y when x = 2, so
[tex]y=16(\frac{2}{3})^2[/tex] and
[tex]y=16(\frac{4}{9})[/tex] and
[tex]y=\frac{64}{9}[/tex] so
y = 7.1 feet
Resorts-R-Us charges $ 125 a night to rent a suite. If you purchase their oneyear membership fee for $350, you only pay $75 a night. Which is a better deal?
Answer:
Step-by-step explanation:
Let x represent the number of nights for which you rent a suite.
Resorts-R-Us charges $ 125 a night to rent a suite. If you choose this plan, the total cost of renting the suite for x nights would be
125 × x = 125x
If you purchase their one year membership fee for $350, you only pay $75 a night. This means that the total cost of renting a suite for x nights would be
75x + 350
For the plan involving membership to be a better deal,
75x + 350 < 125x
350 < 125x - 75x
350 < 50x
50x > 350
x > 350/50
x > 7
After 7 nights, the membership option would be a better deal. If you intend to rent the suite for lesser than 7 nights in a year, then the first option is a better deal.
A study found that a driver’s reaction time A(x) to audio stimuli and his or her reaction time V(x) to visual stimuli (both in milliseconds) can be modeled by
A(x) = 0.0051x² - 0.319x + 15, 16 ≤ x ≤ 70,
V(x) = 0.005x² - 0.23x + 22, 16 ≤ x ≤ 70
where x is the driver's age (in years). Write an inequality that you can use to find the x-values for which A(x) is less than V(x).
Answer:
The required inequality is [tex]0.0001 x^2 - 0.089 x - 7<0[/tex].
Step-by-step explanation:
The given inequalities are
[tex]A(x) = 0.0051x^2 - 0.319x + 15[/tex]
[tex]V(x)= 0.005x^2 - 0.23x + 22[/tex]
where, x is the driver's age (in years), A(x) is driver’s reaction time to audio stimuli and V(x) is his or her reaction time to visual stimuli, 16 ≤ x ≤ 70.
We need to find an inequality that can be use to find the x-values for which A(x) is less than V(x).
[tex]A(x)<V(x)[/tex]
[tex]0.0051x^2 - 0.319x + 15< 0.005x^2 - 0.23x + 22[/tex]
[tex]0.0051x^2 - 0.319x + 15- 0.005x^2 + 0.23x- 22<0[/tex]
Combine like terms.
[tex]0.0001 x^2 - 0.089 x - 7<0[/tex]
where, 16 ≤ x ≤ 70.
Therefore, the required inequality is [tex]0.0001 x^2 - 0.089 x - 7<0[/tex].
Donata bought 3 Apples and 5 Pomegranites at the local supermarket for a total of $16.50 Meaghan bought 6 Apples and 11 Pomegranites at the same store for a total of $35.70 How much does one Apple cost?
Answer: the cost of one apple is $1
Step-by-step explanation:
Let x represent the cost of one apple.
Let y represent the cost of one Pomegranate.
Donata bought 3 Apples and 5 Pomegranates at the local supermarket for a total of $16.50. This means that
3x + 5y = 16.5 - - - - - - - - - - - - -1
Meaghan bought 6 Apples and 11 Pomegranates at the same store for a total of $35.70. This means that
6x + 11y = 35.7- - - - - - - - - - - - -2
Multiplying equation 1 by 2 and equation 2 by 1, it becomes
6x + 10y = 33
6x + 11y = 35.7
Subtracting, it becomes
- y = - 2.7
y = 2.7
Substituting y = 2.7 into equation 1, it becomes
3x + 5 × 2.7 = 16.5
3x + 13.5 = 16.5
3x = 16.5 - 13.5 = 3
x = 3/3 = 1
Determine, in degrees, the measure of each interior angle of a regular octagon
Please show all work on how you got your answer
Answer:
Each interior angle = 135 degrees,
Step-by-step explanation:
The exterior angles of all convex polygons add up to 360 degrees.
So for a regular octagon each exterior angle = 360 / 8
= 45 degrees.
Therefore each interior angle = 180 - 45 = 135 degrees,
A variable that is systematically linked with the factor you believe is causing the overall effect in your research is called the ________ variable. The presence of such a variable can prevent you from knowing what is really causing the effect.
Answer: Extraneous variable
Step-by-step explanation:
In an experiment , Independent variable can be manipulated by the experimenter to see the change in the dependent variable or response variable. But there are some variable known as extraneous variable that is attached to independent variable and make experimenter confuse.
Extraneous variables is defined as :
A variable that is not intentionally involved in any study.It is systematically linked with the independent variable.It can cause effect in research.∴ A variable that is systematically linked with the factor you believe is causing the overall effect in your research is called the extraneous variable variable. The presence of such a variable can prevent you from knowing what is really causing the effect.
Jim's work evaluating 2 (three-fifths) cubed is shown below. 2 (three-fifths) cubed = 2 (StartFraction 3 cubed Over 5 EndFraction) = 2 (StartFraction 3 times 3 times 3 Over 5 EndFraction) = 2 (StartFraction 27 Over 5 EndFraction) = StartFraction 54 Over 5 EndFraction Which statement best describe Jim's first error? He did not multiply Three-fifths by 2 before applying the power. He did not apply the power to the denominator of Three-fifths. He did not evaluate 33 correctly. He did not multiply StartFraction 27 Over 5 EndFraction by 2 correctly.
Jim's first error occurred when he incorrectly applied the cube to three-fifths. He made the mistake of not applying the power to the denominator of the fraction. The answer is 54/125
Explanation:The question revolves around a mathematical operation where Jim is attempting to determine the cube of 2 times three-fifths. Jim's first error lies in how he applied the power of three. According to the rules of exponentiation for fractions, you should apply the power to both the numerator and denominator. Hence, Jim's first mistake was that he did not apply the power to the denominator of three-fifths. Instead of merely cubing the numerator (3), Jim should have also cubed the denominator (5). He should have performed the calculation as follows:
= 2 x (3/5)^3
= 2 x (3^3/5^3)
= 2 (27/125).
= 54/125
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Jim first errored by failing to apply the exponent to both the numerator and denominator of the fraction before multiplying by two. The correct calculation for cubing a fraction should involve cubing both the numerator and the denominator, which would have resulted in 54/125 instead of 54/5.
The first error that Jim made was not applying the power to both the numerator and the denominator of the fraction before multiplying by two. When cubing a number in fractional form, such as three-fifths, one must cube both the numerator (33) and the denominator (53) to correctly apply the exponent to the entire fraction. The correct method is to first cube three-fifths, which results in 27/125, and then multiply that result by 2, leading to the final answer of 54/125, not 54/5 as Jim calculated.
It's important to follow the correct order of operations and apply exponents before multiplication. In this case, to find the value of 2 × (three-fifths) cubed, the calculation should be 2 × (3/5)3, which simplifies to 2 × (27/125), and then to 54/125.
HELP I NEEEEED ANSWER PLEASEEEEE
Answer:
6
Step-by-step explanation:
8^2*(TS)^2=10^2
(TS)^2=36
TS=6
Answer:
Step-by-step explanation:
Triangle RST is a right angle triangle.
From the given right angle triangle,
RS represents the hypotenuse of the right angle triangle.
With m∠R as the reference angle,
RT represents the adjacent side of the right angle triangle.
ST represents the opposite side of the right angle triangle.
To determine m∠R, we would apply
the cosine trigonometric ratio.
Cos θ = adjacent side/hypotenuse. Therefore,
Cos m∠R = 8/10 = 0.8
m∠R = Cos^-1(0.8)
m∠R = 36.9 to one decimal place.
Sam eats 1/2 of a chocolate bar. Jack eats 1/4 of the remaining chocolate bar. After both Jack and Sam eat their portions of the chocolate bar how much is left uneaten of the original whole chocolate bar?
Final answer:
After Sam eats 1/2 and Jack eats 1/4 of the remaining chocolate bar, 3/8 of the original chocolate bar is left uneaten.
Explanation:
Sam eats 1/2 of a chocolate bar. Jack then eats 1/4 of the remaining chocolate bar. To determine how much of the chocolate bar is left, we need to perform a couple of simple calculations. After Sam eats his portion, 1/2 of the bar is left. Jack eats 1/4 of what remains after Sam, which is 1/4 of 1/2, or 1/2 * 1/4 = 1/8 of the original bar.
This means that Jack's portion is 1/8 of the entire chocolate bar. Now, the total amount eaten by Sam and Jack together is 1/2 (Sam's portion) + 1/8 (Jack's portion). To add these fractions, they need to have a common denominator, which in this case is 8. This makes Sam's portion 4/8 when expressed with the common denominator. Now, add 4/8 (Sam's adjusted portion) + 1/8 (Jack's portion) = 5/8.
Therefore, the total amount eaten by both is 5/8 of the bar, leaving 3/8 of the original chocolate bar uneaten.