Answer:
Maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$
Step-by-step explanation:
We are given the following information:[tex]P(x) = 6400x - 18x^2 - \frac{x^3}{3} - 40000[/tex], where P(x) is the profit function.
We will use double derivative test to find maximum profit.
Differentiating P(x) with respect to x and equating to zero, we get,
[tex]\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2[/tex]
Equating it to zero we get,
[tex]x^2 + 36x - 6400 = 0[/tex]
We use the quadratic formula to find the values of x:
[tex]x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex], where a, b and c are coefficients of [tex]x^2, x^1 , x^0[/tex] respectively.
Putting these value we get x = -100, 64
Now, again differentiating
[tex]\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x[/tex]
At x = 64, [tex]\displaystyle\frac{d^2(P(x))}{dx^2} < 0[/tex]
Hence, maxima occurs at x = 64.
Therefore, maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$
Write a complete mathematical definition of (a,b), the greatest common divisor of a,b in \mathbb{Z}
Answer:
The greatest common divisor of two integers a and b (not both 0) is the largest integer that divides both a and b.
Step-by-step explanation:
Think for example of the numbers a=5, and b= -10. The greatest common divisor of 5 and -10, is the largest integer that divides both 5 and -10. We can find it by inspection (although there are more advanced methods to find it). We can list all integers that divide both 5 and -10.
-5 divides 5, and it also divides -10
-1 divides 5, and it also divides -10
1 divides 5, and it also divides -10
5 divides 5, and it also dividies -10
The LARGEST of them all is then 5, so 5 is the greatest common divisor of 5 and -10. The usual way to write it is
[tex]gcd(5,-10)=(5,-10)=5[/tex]
The greatest common divisor (GCD) of two integers a and b is the largest integer that divides both a and b without leaving a remainder. The GCD, denoted as (a, b), can be calculated using the Euclidean algorithm.
The greatest common divisor (GCD) of two non-zero integers a and b, denoted as (a, b), is the largest positive integer that divides both a and b without leaving a remainder. A formal definition could be formulated as follows: The greatest common divisor of a and b is a number d such that:
d is a divisor of a (i.e., a mod d = 0).
d is a divisor of b (i.e., b mod d = 0).
For any other integer e that divides both a and b, e \\leq d.
Moreover, the GCD can be calculated using algorithms such as the Euclidean algorithm, which is based on the principle that the gcd of a and b is the same as the gcd of b and a mod b, assuming a > b and b \\neq 0.
please answer 1 and 2 and if you can explain!!! i need help asap i’ll mark brainliest!!!
Answer:
1) A.
2) No
Step-by-step explanation:
1 is A because the shaded line extends over numbers greater than -8 but less than -1. The reason it is greater than or EQUAL to -1 is because the dot above -1 is shaded in.
2 is No because in order to solve this equation, you plug in the numbers from the coordinates into the inequality. An ordered pair is always structured as (x,y), so in this case x = -1 and y = 4. To solve, the first step is to plug the numbers in, and you end up with 4< 2(-1) +5.
Then, simplify by adding and multiplying as needed. Now you will end up with 4<-2 +5. Simplify again. Finally you end up with 4<3. The reason the answer is NO, not a solution is because the statement '4<3' (four is less than three) is false. if the equation had ended up being 4>3, then it would have been true.
Given the following prescription formula, what is the ratio strength (nearest whole number) of methylcellulose in the finished product? As a ratio is typically expressed as 1:some number, put ONLY the number in the space provided and NOT the 1: portion. DO NOT include any units. • Progesterone 3.8 g • Glycerin 7 mL • 2% methylcellulose solution 50 mL • Cherry syrup ad 90 mL Your Answer: Answer
Answer:
147
Step-by-step explanation:
Given:
Progesterone = 3.8 g
Glycerin = 7 mL
2% methylcellulose solution 50 mL
Cherry syrup ad = 90 mL
Now,
The total volume of the solution = 7 + 50 + 90 = 147 mL
Also,
2% methylcellulose solution 50 mL is concluded as:
the volume of methylcellulose in the solution is 2% of the total volume of the solution
thus,
volume of methylcellulose = 0.02 × 50 mL = 1 mL
Therefore,
Ratio strength of methylcellulose in the finished product
=[tex]\frac{\textup{volume of methylcellulose}}{\textup{ total volume of the solution}}[/tex]
or
= [tex]\frac{\etxtup{1}}{\textup{ 147}}[/tex]
Hence, the answer according to the question is 147
The clutch linkage on a vehicle has an overall advantage of 24:1. If the pressure plate applies a force of 504lb,how much force must the driver apply to release the clutch.
Answer:
the force that must be applied by the driver to release the clutch is 21 lb
Step-by-step explanation:
Data provided:
clutch linkage on a vehicle has an overall advantage = 24:1
Applied force by the pressure plate = 504 lb
Now,
the advantage ratio is given as:
advantage ratio = [tex]\frac{\textup{Force applied by the pressure plate}}{\textup{Force applied by the driver}}[/tex]
on substituting the respective values, we get
[tex]\frac{\textup{24}}{\textup{1}}[/tex] = [tex]\frac{\textup{504 lb}}{\textup{Force applied by the driver}}[/tex]
or
Force applied by the driver to release the clutch = [tex]\frac{\textup{504 lb}}{\textup24}}[/tex]
or
Force applied by the driver to release the clutch = 21 lb
Hence,
the force that must be applied by the driver to release the clutch is 21 lb
Using the mechanical advantage of the clutch linkage (24:1), the force the driver must apply to release the clutch is calculated to be 21 pounds.
The student has asked about the amount of force a driver must apply to release the clutch in a vehicle, given that the clutch linkage has an overall mechanical advantage of 24:1 and the pressure plate applies a force of 504lb. To find the force the driver needs to apply (Fdriver), we use the relationship provided by the mechanical advantage. Mechanical advantage (MA) is defined as the output force (Fout) divided by the input force (Fdriver). From this, we can formulate the equation MA = Fout / Fdriver, which can be rearranged to solve for the driver's force: Fdriver = Fout / MA.
Substituting the given values:
Fdriver = 504lb / 24
Fdriver = 21lb
Therefore, the driver must apply a force of 21 pounds to release the clutch.
In a recent poll, 850 people were surveyed. 69% of them said they opposed the current war. How many people stated they were in opposition to the war?
Answer:
586 people stated that they were in opposition to the war.
Step-by-step explanation:
Percentage problems can be solved as a simple rule of three problem:
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.
A percentage problem is an example where the relationship between the measures is direct.
The problem states that 69% of the said they opposed the current war. 69% is 69 of 100. How much it is of 850.
So
69 - 100
x - 850
100x = 69*850
[tex]x = \frac{58650}{100}[/tex]
x = 586.5
586 people stated that they were in opposition to the war.
A new security system needs to be evaluated in the airport. The probability of a person being a security hazard is 0.02. At the checkpoint, the security system denied a person without security problems 1.5% of the time. Also the security system passed a person with security problems 1% of the time. What is the probability that a random person does not pass through the system and is without any security problems? Report answer to 3 decimal places.
Final answer:
Using the given probabilities, we find that the probability is 0.965, or 96.5%.
Explanation:
To find the probability that a random person does not pass through the system and is without any security problems, we need to calculate the complement of two events: a person being a security hazard and the security system denying a person without security problems.
First, let's calculate the probability of a person being a security hazard:
Probability of a person being a security hazard = 0.02
Next, let's calculate the probability of the security system denying a person without security problems:
Probability of the security system denying a person without security problems = 1.5% = 0.015
To find the probability that a person does not pass through the system and is without any security problems, we can use the formula:
Probability = (1 - probability of being a security hazard) * (1 - probability of the security system denying a person without security problems)
Probability = (1 - 0.02) * (1 - 0.015)
Probability = 0.98 * 0.985
Probability = 0.9653
Therefore, the probability that a random person does not pass through the system and is without any security problems is 0.965, or 96.5% (rounded to 3 decimal places).
Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 1% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent?
Answer:
the probability that the user is fraudulent is 0.00299133
Step-by-step explanation:
Let be the events be:
G: The user generates calls from two or more areas.
NG: The user does NOT generate calls from two or more areas.
L: The user is legitimate.
F: The user is fraudulent.
The probabilities established in the statement are:
[tex]P (G | L) = 0.01//P (G | F) = 0.30//P (F) = 0.0001//P (L) = 0.9999//[/tex]
With these values, the probability that a user is fraudulent, if it has originated calls from two or more areas is:
[tex]P (F|G) = \frac{P(F\bigcap G)}{P(G)} = \frac{P(F)P(G|F)}{P(G)} = \frac{P(F)P(G|F)}{P(F)P(G|F)+P(L)P(G|L)}[/tex]
[tex]\frac{(0.0001)(0.30)}{(0.0001)(0.30)+(0.9999)(0.01)} = 0.00299133[/tex]
In Triangle ABC, angle A is congruent to angle B and
themeasure of angle A is 54. Find the measure of angle C, the value
ofAC and BC.
Answer:
Angle C=72º
Step-by-step explanation:
If two angles are congruent they are equals then angle A is 54º and angle B is 54º and the sum of the internal angles of a triangle is 180º then.
C=180º-54º-54º=72º
To know the value of on AC and BC we have to know the value of the other side AB, to find the values of the sides we can use the law of sines;
[tex]\dfrac{AB}{sin(72)}=\dfrac{BC}{sin(54)}=\dfrac{AC}{sin(54)}[/tex]
The owner of the Rancho Los Feliz has 7000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. Fencing is not required along the river, and the length of the fencing parallel to the river is to exceed the length of the fencing perpendicular to it by 2500 yd. Find the area of the enclosed land (in sq yd).
Answer:
6000000 sq yd
Step-by-step explanation:
Data provided in the question:
Length of the fencing = 7000 yd
let the perpendicular sides be 'P'
and the length parallel to the river be 'L'
according to the given question
L = P + 2500 ............(1)
also,
Length to be fenced = 2P + L
thus,
2P + L = 7000 ...........(2)
substituting L from (1), we get
2P + P + 2500 = 7000
or
3P = 7000 - 2500
or
3P = 4500
or
P = 1500 yd
Thus,
L = 1500 + 2500 = 4000 yd
Therefore,
the area of the rectangular land = L × P = 4000 × 1500 = 6000000 sq yd
Answer:
Area of land = 6000000 sq yd
Step-by-step explanation:
Given,
length of fencing= 7000 yd
Let's assume that the length of the land parallel to the river is l and the breadth of the land perpendicular to the river is b.
Then, it is given that
l = b +2500
Since, there is no need of fencing along the river so, we can write
l +2b = 7000
=>b+2500 = 7000
=> b = 7000-2500
= 4000
As the area of rectangular land can be given as
A = length x breadth
= 4000 x 2500 sq yd
= 6000000 sq yd
So, the area of the enclosed land will be 6000000 sq yd.
linear equations: 1.5x + 1.3x= -8.4
1.5x + 1.3x = - 8.4
2.8x = -8.4
x= - 8.4/2.8
x = 3
YOURE WELCOME PLEASE MARK ME AS BRAINLIEST AND FOLLOW ME ❤️❤️❤️❤️❤️ AU REVOIROn simplification of liner equation 1.5x + 1.3x = -8, we get x = -3.
To solve the linear equation 1.5x + 1.3x = -8.4, we need to start by combining like terms.
Both terms on the left side of the equation have the variable x, so we can add them together.
1.5x + 1.3x = 2.8x.
2.8x = -8.4.
To find the value of x, we need to isolate the variable by dividing both sides of the equation by 2.8.
Divide both sides by 2.8:
x = -8.4 / 2.8.
x = -3.
Plz Help if possible
Write the equation of the line, in standard form, that has a y-intercept of 2 and is parallel to 2x + y = -5. Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
Answer:
2x + y = 2.
Step-by-step explanation:
First find the slope of the required line by writing the line 2x + y = -5 in slope intercept form:
2x + y = -5
y = -2x - 5
- so the slope is -2.
Therefore the required equation is
y = -2x + 2 (where 2 is the y-intercept).
Converting to standard form:
y = -2x + 2
2x + y = 2.
The equation of the line parallel to 2x + y = -5 with a y-intercept of 2 is 2x + y = 2.
Explanation:To find the equation of a line parallel to the given line, we must first realize that parallel lines have the same slope. The given equation is 2x + y = -5, which can be rearranged into y = -2x - 5, showing us that the slope of the given line is -2. Therefore, the slope of the line we want to find is also -2.
With a slope of -2 and a y-intercept of 2 (since the line is said to intersect with the y-axis at y=2), the slope-intercept form of the line is y = -2x + 2. However, the question requires the answer in standard form. The standard form is Ax + By = C, where A, B and C are integers and A > 0. To convert our slope-intercept equation to standard form, we will add 2x to both sides, obtaining the final equation as 2x + y = 2.
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Convert 17.42 m to customary units. A.57'-17/8" B. 36-10 1/2" C. 442 1/2" D. 367/8" E. None of these answers is reasonable.
Answer:
Option E - None of these answers is reasonable.
Step-by-step explanation:
To find : Convert 17.42 m to customary units ?
Solution :
The customary units is defined as the measure length and distances in the customary system are inches, feet, yards, and miles.
The options belong to feet and inches.
We have to convert meter into inches, feet.
Meter into feet,
[tex]1 \text{ feet} = 0.3048 \text{ meter}[/tex]
[tex]1 \text{ meter} = \frac{1}{0.3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} = \frac{17.42}{0.3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} = \frac{174200}{3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} =57 \frac{464}{3048}\text{ feet}[/tex]
[tex]17.42 \text{ meter} =57 \frac{58}{381}\text{ feet}[/tex]
Now, Feet into inches
[tex]1 \text{ feet} = 12\text{ inches}[/tex]
[tex] \frac{58}{381} \text{ feet} = 12\times \frac{58}{381}\text{ inches}[/tex]
[tex] \frac{58}{381} \text{ feet} =\frac{232}{381}\text{ inches}[/tex]
i.e. [tex]17.42 \text{ meter} =57\text{ feet }\frac{232}{381}\text{ inches}[/tex]
or [tex]17.42 \text{ meter} =57'\frac{232}{381}''[/tex]
None of these answers is reasonable.
Therefore, Option E is correct.
Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection. Give your answers in degrees, rounding to one decimal place. Enter your answers as a comma-separated list.) y = 7x2, y = 7x3
Answer:
The angles between the curves at the points of intersection are:
0º, 1.3º
Step-by-step explanation:
The intersections points are found by setting the equations equal to each other and solving the resulting equation:
[tex]7x^2=7x^3\\x^3-x^2=0\\x^2(x-1)=0\\x=0,x=1[/tex]
The angles of the tangent lines can be found by stating their slopes.
To find the slope we differentiate the equations:
[tex]y'_1=14x,y'_2=21x^2[/tex]
Then we plug the x-coordinates of the intersections:
For x=0 we get the slopes are both 0:
[tex]y'_1=14(0)=0,y'_2=21(0)^2=0[/tex]
So the angles of inclination of the lines are the same their difference is 0. Hence the angle between the tangent curves is also 0º at the point of intersection at x=0
For x=1 we get the following slopes:
[tex]y'_1=14(1)=14,y'_2=21(1)^2=21[/tex]
The slopes are the tangents of the angles. Therefore, to get the angle between the lines we do:
[tex]arctan(21)-arctan(14)\approx87.2737\º-85.9144\º\approx1.3\º[/tex]
So, 1.3º is the angle between the curves at the second point of intersection at x=1.
Nancy thinks the answer to the question 1/4 divided by -2/3 is 3/8 is next correct explain why or why not be sure to use complete sentences
Answer:
Her sign is in error. The answer is -3/8.
Step-by-step explanation:
Nancy's answer has the correct magnitude. It is obtained by multiplying 1/4 by -3/2. However, the sign of that product will be negative. Nancy has reported a positive answer, so it is incorrect.
Directions: Use system of linear equations
9x-3y=-1
1/5x+2/5y=-1/3
Answer:
The solution to this set of linear equations is:
[tex]x=-\frac{1}{3}\\y=-\frac{2}{3}[/tex]
Step-by-step explanation:
This is a system of two equations with two unknown variables x and y, let's call them
Equation 1: [tex]9x-3y=-1[/tex]
Equation 2: [tex]\frac{1}{5}x+\frac{2}{5}y=-\frac{1}{3}[/tex]
The first step is to solve Equation 1 for y, this means to leave the y alone on one side of the equal
[tex]y= 3x+\frac{1}{3}[/tex]
Then with this equation, you can find the value of x by replacing y in Equation 2
[tex]\frac{1}{5}x+\frac{2}{5}(3x+\frac{1}{3})=-\frac{1}{3}[/tex]
Then simplify this equation to find x
[tex]\frac{1}{5}x+\frac{6}{5}x+\frac{2}{15}=-\frac{1}{3}[/tex]
[tex]\frac{1}{5}x+\frac{6}{5}x=-\frac{1}{3}-\frac{2}{15}[/tex]
[tex]\frac{7}{5}x=-\frac{5}{15}-\frac{2}{15}[/tex]
[tex]\frac{7}{5}x=-\frac{7}{15}[/tex]
Now you solve for x
[tex]x=-\frac{1}{3}[/tex]
Now you use this value of x to find y
[tex]y=3(-\frac{1}{3})+\frac{1}{3}\\y=-\frac{2}{3}[/tex]
You can check if this answer is correct by replacing the values of x and y into Equation 1 or 2, in this case, let's take Equation 1:
[tex]9(-\frac{1}{3})-3(-\frac{2}{3})=-1\\-3+2=-1\\-1=-1\\[/tex]
If a population of dolphins increases at a constant rate of 1.5% every year for 20 years what will be the total percentage increase over the 20 years.
Answer:
The population increased by 34.69% over 20 years.
Step-by-step explanation:
It is given that the population of dolphins increases at a constant rate of 1.5% every year for 20 years.
Formula for population increase:
[tex]P=a(1+r)^t[/tex]
where, a is initial population, r is growth rate and t is time in years.
If the population of dolphins increases at a constant rate of 1.5% every year for 20 years, then the population after 20 years is
[tex]P=a(1+0.015)^{20}[/tex]
[tex]P=a(1.015)^{20}[/tex]
[tex]P=1.346855a[/tex]
Where, a is the initial population.
The total percentage increase over the 20 years is
[tex]\% change=\frac{P-a}{a}\times 100[/tex]
where, P is population after 20 years and a is initial amount.
[tex]\% change=\frac{1.346855a-a}{a}\times 100[/tex]
[tex]\% change=\frac{0.346855a}{a}\times 100[/tex]
[tex]\% change=0.346855\times 100[/tex]
[tex]\% change=34.6855[/tex]
[tex]\% change\approx 34.69[/tex]
Therefore the population increased by 34.69% over 20 years.
A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is 1.20 m above the pool. (a) How long are her feet in the air? .3 Incorrect: Your answer is incorrect. s (b) What is her highest point above the board? m (c) What is her velocity when her feet hit the water? m/s
The height of the swimmer's feet in the air at time [tex]t[/tex] is given according to
[tex]y=1.20\,\mathrm m+\left(4.00\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2[/tex]
where [tex]g[/tex] is the magnitude of the acceleration due to gravity (taken here to be 9.80 m/s^2).
a. Solve for [tex]t[/tex] when [tex]y=0[/tex]:
[tex]1.20\,\mathrm m+\left(4.00\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2=0\implies\boxed{t=1.05\,\mathrm s}[/tex]
(The other solution is negative; ignore it)
b. At her highest point [tex]y_{\rm max}[/tex], the swimmer has zero velocity, so
[tex]-\left(4.00\dfrac{\rm m}{\rm s}\right)^2=-2g(y_{\rm max}-1.20\,\mathrm m)\implies\boxed{y_{\rm max}=2.02\,\mathrm m}[/tex]
c. Her velocity at time [tex]t[/tex] is
[tex]v=4.00\dfrac{\rm m}{\rm s}-gt[/tex]
After 1.05 s in the air, her velocity will be
[tex]v=4.00\dfrac{\rm m}{\rm s}-g(1.05\,\mathrm s)\implies\boxed{v=-6.29\dfrac{\rm m}{\rm s}}[/tex]
The swimmer's feet are in the air for approximately 0.816 seconds. Her highest point above the diving board is approximately .43 m. She hits the water with a velocity of approximately -8.00 m/s.
Explanation:To answer these questions, we need to use physics equations that describe motion. The swimmer's motion can be broken down into two parts - the upward motion and the downward motion. Let's discuss each with respect to the provided variables.
(a) How long are her feet in the air?
To calculate the time, we can use the equation of motion given by: t = (v_f - v_i)/g where v_f is the final velocity (which is 0 at the highest point), v_i is the initial velocity (4.00 m/s), and g is the acceleration due to gravity (approx -9.81m/s²). The time taken for the upwards journey is: t = (0 - 4)/-9.81 ≈ 0.408 seconds. Since motion up and motion down take the same amount of time, we double this to get the total time: 2*0.408 = 0.816 seconds.
(b) What is her highest point above the board?
Let's use the equation h = v_i * t + 0.5*g*t², where h is the height, t is the time (0.408 seconds), g is the gravity (-9.81 m/s²), and v_i is the initial velocity (4.00 m/s). The highest point above the board is: h = 4*0.408 + 0.5*-9.81* (0.408)² = 1.63 m above the water surface or .43 m above the diving board.
(c) What is her velocity when her feet hit the water?
Here, we can repurpose the equation v_f = v_i + g*t. Notice that the time here is the total time her feet were in the air (0.816 seconds). Using these values we get: v_f = 0 + (-9.81 * 0.816) = -8.00 m/s. She hits the water at a speed of 8.00 m/s.
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Convert these decimals into fractions and simplify as much as possible: a) 0.25 b) 0.08 C) 0.400 d) 1.1 e) 3.5
Answer:
When we change a decimal into fraction, then we follow the following steps,
Step 1 : first we write the decimal number with the denominator 1,
Step 2 : Multiply numerator by 10s ( eg 10, 100, 100 etc ) for omitting decimal.
Step 3 : Multiply the denominator ( i.e 1 ) by the same number,
Step 4 : Reduce the fraction in the simplest form if possible by dividing both numerator and denominator by the HCF of numerator and denominator.
a)
[tex]0.25 =\frac{0.25}{1}=\frac{0.25\times 100}{100}=\frac{25}{100}=\frac{25\div 25}{100\div 25}=\frac{1}{4}[/tex]
b)
[tex]0.08 =\frac{0.08}{1}=\frac{0.08\times 100}{100}=\frac{8}{100}=\frac{8\div 4}{100\div 4}=\frac{2}{25}[/tex]
c)
[tex]0.400 =\frac{0.4}{1}=\frac{0.4\times 10}{10}=\frac{4}{10}=\frac{4\div 2}{10\div 2}=\frac{2}{5}[/tex]
d)
[tex]1.1 =\frac{1.1}{1}=\frac{1.1\times 10}{10}=\frac{11}{10}[/tex]
e)
[tex]3.5 =\frac{3.5}{1}=\frac{3.5\times 10}{10}=\frac{35}{10}=\frac{35\div 5}{10\div 5}=\frac{7}{2}[/tex]
Final answer:
To convert decimals into simplified fractions: 0.25 is 1/4, 0.08 is 2/25, 0.400 is 2/5, 1.1 is 11/10 and 3.5 is 7/2. Numbers in scientific notation are written in decimal form by adjusting the decimal point. When rounding to three significant figures, ensure only the first three digits after the leading non-zero digit are kept.
Explanation:
When converting decimals to fractions and simplifying them, it's important to consider the place value of the decimal. Here's how you would convert and simplify the provided decimals:
0.25 can be written as 25/100, which simplifies to 1/4.
0.08 is 8/100, which simplifies to 1/12.5 or 2/25 when expressed as a simplified fraction.
0.400 is 400/1000, which simplifies to 2/5.
1.1 is equivalent to 11/10 or 1 1/10 in mixed number form.
3.5 equals 35/10, which simplifies to 7/2 or 3 1/2 in mixed number form.
For scientific notation, numbers are converted to their decimal forms by moving the decimal point:
5.65 x 10-3 means the decimal point is moved 3 places to the left, giving 0.00565.
9.25 x 10-4 means the decimal point is moved 4 places to the left, resulting in 0.000925.
To write numbers in scientific notation:
4500 becomes 4.5 x 103.
2220000 turns into 2.22 x 106.
0.0035 is 3.5 x 10-3.
0.7 can be written as 7 x 10-1.
858.67 is expressed as 8.5867 x 102.
When rounding to three significant figures:
0.0004505 becomes 4.51 x 10-4 (count starts from the first non-zero digit).
0.00045050 also rounds to 4.51 x 10-4.
For 7.210 x 106, it remains unchanged as it already has three significant figures.
5.00 x 10-6 stays the same, with three significant figures present.
prove that |a| < b if and only if -b < a < b
Answer:
Since,
[tex]|x|=\left\{\begin{matrix}x &\text{ if } x \geq 0 \\ -x &\text{ if } x < 0\end{matrix}\right.[/tex]
Here, the given equation is,
|a| < b
Case 1 : if a ≥ 0,
|a| < b ⇒ a < b
Case 2 : If a < 0,
|a| < b ⇒ -a < b ⇒ a > - b
( Since, when we multiply both sides of inequality by negative number then the sign of inequality is reversed. )
|a| < b ⇒ a < b or a > - b ⇒ -b < a < b
Conversely,
If -b < a < b
⇒ a < b or a > - b
⇒ a < b or -a < b
⇒ |a| < b
Hence, proved..
Using data collected between 1957 and 1978, from 15 samples of adults, Professor Rodgers (1982 Social Forces) found that (1) the average level of happiness reported by people under 65 years of age declined from 1957 to 1970. For this same group, the average level of happiness increased slightly from 1970 to 1978. (2) The average level of happiness reported by people age 65 and older increased from 1957 to 1978. A (the) variable(s) in this study is (are)
Answer:
The variable in this study is age.
Step-by-step explanation:
The variable in this study is Age, which has a relationship of cause and effect. Consequently,it is clear that happiness does not depend on the passing of time , but on the age of each group of people.
A = ( −2 −1 2 −2 2 3 −4 1 3 ) b = ( −1 −1 4 ) x = ( x1 x2 x3 ) (a) (2 pts) Write down the augmented matrix (A|b). (b) (4 pts) Use Gauss-Jordan elimination to find the Reduced Row Echelon Form (RREF) of the augmented matrix. (c) (2 pts) What is the rank of A? What is the rank of (A|b) (d) (2 pts) State whether the system is consistent or inconsistent. State how many solutions the system has; if there is/are a solution/s, write it/them down.
Answer:
The augmented matrix is [tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
The Reduced Row Echelon Form of the augmented matrix is [tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
The rank of matrix (A|B) is 3
The system is consistent and the solutions are [tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]
Step-by-step explanation:
We have the following information:
[tex]A=\left[\begin{array}{ccc}-2&-1&2\\-2&2&3\\-4&1&3\end{array}\right], X=\left[\begin{array}{c}x_{1}&x_{2}&x_{3}\end{array}\right] and \:B=\left[\begin{array}{c}-1&-1&4\end{array}\right][/tex]
1. The augmented matrix is
We take the matrix A and we add the matrix B we use a vertical line to separate the coefficient entries from the constants.
[tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
2. To transform the augmented matrix to the Reduced Row Echelon Form (RREF) you need to follow these steps:
Row operation 1: multiply the 1st row by -1/2[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]
Row Operation 2: add 2 times the 1st row to the 2nd row[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\-4&1&3&4\end{array}\right][/tex]
Row Operation 3: add 4 times the 1st row to the 3rd row[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\0&3&-1&6\end{array}\right][/tex]
Row Operation 4: multiply the 2nd row by 1/3[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&3&-1&6\end{array}\right][/tex]
Row Operation 5: add -3 times the 2nd row to the 3rd row[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&-2&6\end{array}\right][/tex]
Row Operation 6: multiply the 3rd row by -1/2[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&1&-3\end{array}\right][/tex]
Row Operation 7: add -1/3 times the 3rd row to the 2nd row[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
Row Operation 8: add 1 times the 3rd row to the 1st row[tex]\left[\begin{array}{cccc}1&1/2&0&-5/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
Row Operation 9: add -1/2 times the 2nd row to the 1st row[tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]
3. What is the rank of (A|B)
To find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.
Because the row echelon form of the augmented matrix has three non-zero rows the rank of matrix (A|B) is 3
4. Solutions of the system
This definition is very important: "A system of linear equations is called inconsistent if it has no solutions. A system which has a solution is called consistent"
This system is consistent because from the row echelon form of the augmented matrix we find that the solutions are (the last column of a row echelon form matrix always give you the solution of the system)
[tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]
Consider the area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant. (a) Set up the definite integral needed to find the area. (b) Make an appropriate substitution
[tex]y(t)=8\sin(bt)[/tex] has a period of [tex]\dfrac{2\pi}b[/tex], which is to say one "arch" of the curve occurs over the interval [tex]0\le t\le\dfrac\pi b[/tex].
a. Then the area under one such arch is
[tex]\displaystyle\int_0^{\pi/b}8\sin(bt)\,\mathrm dt[/tex]
b. Substitute [tex]u=bt[/tex], so that [tex]\dfrac{\mathrm du}b=\mathrm dt[/tex]. When [tex]t=0[/tex], [tex]u=0[/tex]; when [tex]t=\dfrac\pi b[/tex], [tex]u=\pi[/tex].
Then the integral is
[tex]\displaystyle\frac1b\int_0^\pi8\sin u\,\mathrm du[/tex]
The required area is [tex]\int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]
Given that,
The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.
We have to find,
Set up the definite integral needed to find the area.
Make an appropriate substitution.
According to the question,
The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.
The curve y(t) = 8sin(bt) has a period of 2π\b, which is one arch of the curve occur over the interval [tex]0\leq t\leq \frac{\pi }{b}[/tex].The area under one arch is given by,
[tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
The required area is [tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]
Appropriate Substitute u= bt ,Then,
[tex]\dfrac{du}{b} = dt \\\\when \ t=0, \ and \ u=0\\\\when\ t = \dfrac{\pi }{b}, u = \pi[/tex]
Then,
The required integral is ,
[tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]
The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex].
To know more about Integration click the link given below.
https://brainly.com/question/17721279
Given the table below.. Find the following..
a) Find formula for h(t)
b) Find t intercept of function as an ordered pair
c) Find h intercept of function as an ordered pair
Answer:
h(t)=-1/3(x)+50/3
h intercept is (0,50/3)
t intercept is (50,0)
Step-by-step explanation:
Find the slope of the table by using the slope formula then plug in to y-y1=m(x-x1) then solve for y this gives you the formula
sub in y =0 for the x intercept
sub in x=0 for the y intercept
A diver's elevation is -5 feet relative to sea level. A
school of fish is swimming at an elevation of -12 feet.
What is the difference in elevation between the diver
and the school of fish?
Answer:
-7 feet
Step-by-step explanation:
To find the difference in elevation between the diver and the school fish SUBTRACT the elevation of the diver from that of the fish
i.e. difference in elevation = -12 - (-5)
= -12 + 5
= -7 feet
Final answer:
The difference in elevation between the diver at -5 feet and the school of fish at -12 feet is 7 feet, calculated by taking the absolute value of their elevations' difference.
Explanation:
The question asks for the difference in elevation between a diver and a school of fish, with the diver at -5 feet and the fish at -12 feet relative to sea level. To find the difference in elevation, you subtract the diver's elevation from the fish's elevation.
Here is the calculation:
School of fish elevation: -12 feetDiver's elevation: -5 feetDifference in elevation: |-12 - (-5)| = |-12 + 5| = |-7| = 7 feetThe absolute value is used because we are interested in the positive difference in elevation, which is the distance between the two elevations regardless of direction.
Therefore, the difference in elevation between the diver and the school of fish is 7 feet.
A(1,3,5, 7,9}, B {0,1,2,3,4,5,6, 7,8,9} be the universal set and let A (0,2,4,6,8}, and C {2,3,5,7}. List the elements of the following sets 1.) Let U (a) An B (b) C\B (c) AU (Bnc)
Answer:
a) [tex]A \cap B = \{\phi\}[/tex]
b)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
c)[tex]\{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
Step-by-step explanation:
We are given the following information:
The universal set is : U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {0, 2, 4, 6, 8}
B = {1, 3, 5, 7, 9}
C = {2, 3, 5, 7}
a) [tex]A \cap B = \{\phi\}[/tex]
b)[tex] C/B = C \cap \overline{B} = \{2, 3, 5, 7\} \cap \{0, 2, 4, 6, 8\} = \{2\}[/tex]
c)
[tex]A \cup (B \cap C) = \{0, 2, 4, 6, 8\} \cup (\{1, 3, 5, 7, 9\} \cap \{2, 3, 5, 7\})\\=\{0, 2, 4, 6, 8\} \cup \{3, 5, 7\}\\= \{0, 2, 3, 4, 5,6, 7, 8\}[/tex]
The numbers on two consecutively numbered gym lockers have a sum of
135.
What are the locker numbers?
Answer:
67,68
Step-by-step explanation:
Consecutive numbers are the numbers that follow each other. They may be arranged from the smaller to larger or larger to smaller ones.
Some examples of consecutive numbers are 42,43,45,46,... or 67,68,69,70,...
Now, let x be the smaller gym locker number.
The number consecutive to x will be x+1. Thus the larger gym locker number is x+1.
It is given in the question that the sum of these two locker number is 135.
⇒(x)+(x+1) = 135
⇒ 2x + 1 = 135
⇒ 2x = 134
⇒ x =67
Thus, the two consecutive locker number are x = 67 and x+1 = 68.
Final answer:
The two consecutively numbered gym lockers that add up to 135 are 67 and 68.
Explanation:
The locker numbers are 67 and 68.
To find the locker numbers, we can set up an equation where x represents the smaller locker number. Since the lockers are consecutively numbered, the larger locker number is x+1. The sum of the two locker numbers is x + (x + 1) = 135. By solving this equation, we find that x = 67, making the locker numbers 67 and 68.
11101010100110.0101010111 base 2 (convert this number to base 4)
Answer:
3222212.11113
Step-by-step explanation:
First, you should take care of the fractional separator (the dot) so we split the problem in two parts: one for the integer and other for the fractional part.
Since 4 is a power of 2, we can just take two digits from the orignal number and asign it to its corresponding number in base 4:
[tex]\left[\begin{array}{cc}Binary&Base 4\\00&0\\01&1\\10&2\\11&3\end{array}\right][/tex]
Start with the fractional part from the fractional point to the right:
[tex]\left[\begin{array}{ccccc}01&01&01&01&11\\1&1&1&1&3\end{array}\right][/tex]
Then do the same to the integer part starting from the fractional point to the left.
[tex]\left[\begin{array}{ccccccc}11&10&10&10&10&01&10\\3&2&2&2&2&1&2\end{array}\right][/tex]
By joining them together, we obtain the response.
Let A and B be two events in a sample space S such that P(A) = 0.5, P(B) = 0.6, and P(A intersectionB) = 0.15. Find the probabilities below. Hint: (A intersectionBc) union (A intersectionB) = A.
(a) P(A|Bc)
(b) P(B|Ac)
Answer:
(a) [tex]\frac{7}{8}[/tex]
(b) [tex]\frac{9}{10}[/tex]
Step-by-step explanation:
Given,
P(A) = 0.5 ⇒ [tex]P(A^c)=1-P(A) = 1 - 0.5 = 0.5[/tex]
P(B) = 0.6 ⇒ [tex]P(B^c)=1-P(B) = 1 - 0.6 = 0.4[/tex]
P(A∩B) = 0.15
∵ [tex]P(A\cap B^c)=P(A) - P(A\cap B) = 0.5 - 0.15 = 0.35[/tex]
Similarly,
[tex]P(B\cap A^c)=P(B) - P(B\cap A) = 0.6 - 0.15 = 0.45[/tex]
Now,
(a) [tex]P(\frac{A}{B^c})=\frac{P(A\cap B^c)}{P(B^c)}=\frac{0.35}{0.4}=\frac{35}{40}=\frac{7}{8}[/tex]
(b) [tex]P(\frac{B}{A^c})=\frac{P(B\cap A^c)}{P(A^c)}=\frac{0.45}{0.5}=\frac{45}{50}=\frac{9}{10}[/tex]
Ax = b: A = ( −2 −1 2 −2 2 3 −4 1 3 ) b = ( −1 −1 4 ) x = ( x1 x2 x3 ) (a) (6 pts) Use the elimination method to find the matrices L and U such that A = LU. (b) (3 pts) Solve the intermediate system Ly = b for the intermediate variable y. (c) (3 pts) Solve the system Ux = y and confirm your solution from Problem 2 above
[tex]\underbrace{\begin{bmatrix}-2&-1&2\\-2&2&3\\-4&1&3\end{bmatrix}}_A\underbrace{\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}}_x=\underbrace{\begin{bmatrix}-1\\-1\\4\end{bmatrix}}_b[/tex]
Multiply [tex]A[/tex] on the left side with the following elimination matrix [tex]E_1[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\-1&1&0\\-2&0&1\end{bmatrix}}_{E_1}A=\begin{bmatrix}-2&-1&2\\0&3&1\\0&3&-1\end{bmatrix}[/tex]
Multiply [tex]E_1A[/tex] on the left by another elimination matrix [tex]E_2[/tex]:
[tex]\underbrace{\begin{bmatrix}1&0&0\\0&1&0\\0&-1&1\end{bmatrix}}_{E_2}(E_1A)=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}[/tex]
[tex]\implies\boxed{U=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}}[/tex]
Multiply on the left by the inverse of [tex]E_2E_1[/tex]:
[tex](E_2E_1)^{-1}(E_2E_1)A=(E_2E_1)^{-1}U[/tex]
[tex]A=\underbrace{({E_1}^{-1}{E_2}^{-1})}_LU[/tex]
We have
[tex]{E_1}^{-1}=\begin{bmatrix}1&0&0\\1&1&0\\2&0&1\end{bmatrix}[/tex]
[tex]{E_2}^{-1}=\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix}[/tex]
[tex]\implies\boxed{L=\begin{bmatrix}1&0&0\\1&1&0\\3&1&1\end{bmatrix}}[/tex]
7. When companies employ control charts to monitor the quality of their products, a series of small samples is typically used to determine if the process is "in control" during the period of time in which each sample is selected. Suppose a concrete-block manufacturer samples nine blocks per hour and tests the breaking strength of each. During one-hour’s test, the mean and standard deviation are 985.6 pounds per square inch (psi) and 22.9 psi, respectively. a. Construct a 99% confidence interval for the mean breaking strength of blocks produced
Answer:
A 99% confidence interval for the mean breaking strength of blocks produced is [tex][959.987, 1011.213][/tex]
Step-by-step explanation:
A (1 - [tex]\alpha[/tex])x100% confidence interval for the average break in these conditions It is an interval for the population mean with unknown variance and is given by:
[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]
[tex]\bar X = 985.6psi[/tex]
[tex]n = 9[/tex]
[tex]\alpha = 0.01[/tex]
[tex]T_{(n-1,\frac{\alpha}{2})}=3.355[/tex]
[tex]S = 22.9[/tex]
With this information the interval is determined by:
[tex][985.6 - 3.355\frac{22.9}{\sqrt{9}}, [985.6 - 3.355\frac{22.9}{\sqrt{9}}] = [959.987, 1011.213] [/tex]