Answer:
The original price is = $75
The reduced price = $50
So, price reduced is = [tex]75-50=25[/tex] dollars
Total swimsuits are = [tex]40+25=65[/tex]
Total markdown = [tex]65\times25=1625[/tex] dollars
Now, 25 swimsuits were returned to the original price. Means 25 swimsuits were returned to $75, increasing $25 again.
So, markdown cancellation = [tex]25\times25=625[/tex] dollars
Net markdown = total markdown - markdown cancellation
= [tex]1625-625=1000[/tex] dollars
Find the complementary angle of 73.8 (Type an integer or a decimal.) is Enter your answer in the answer box
Answer:
16.2
Step-by-step explanation:
Since, if the sum of two angles is 90° then they are called complementary angles.
Here, the given angle is 73.8°,
Let x be the complementary angle of 73.8°,
Thus by the above definition,
x + 73.8° = 90°,
By subtraction property of equality,
⇒ x = 16.2°,
Hence, the complementary angle of 73.8° is 16.2°.
Please help me with this
Answer:
m∠1=80°, m∠2=35°, m∠3=33°
Step-by-step explanation:
we know that
The sum of the interior angles of a triangle must be equal to 180 degrees
step 1
Find the measure of angle 1
In the triangle that contain the interior angle 1
∠1+69°+31°=180°
∠1+100°=180°
∠1=180°-100°=80°
step 2
Find the measure of angle 2
In the small triangle that contain the interior angle 2
∠2+45°+(180°-∠1)=180°
substitute the value of angle 1
∠2+45°+(180°-80°)=180°
∠2+45°+(100°)=180°
∠2+145°=180°
∠2=180°-145°=35°
step 3
Find the measure of angle 3
In the larger triangle that contain the interior angle 3
(∠3+31°)+69°+47°=180°
∠3+147°=180°
∠3=180°-147°=33°
Let sin t = a, cos t = b, and tan t = c. Write the expression in terms of a, b, and c.
-sin(-t - 8 π) + cos(-t - 2 π) + tan(-t - 5 π)
Answer:
[tex]a+b-c[/tex]
*Note c could be written as a/b
Step-by-step explanation:
-sin(-t - 8 π) + cos(-t - 2 π) + tan(-t - 5 π)
The identities I'm about to apply:
[tex]\sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)[/tex]
[tex]\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)[/tex]
[tex]\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}[/tex]
Let's apply the difference identities to all three terms:
[tex]-[\sin(-t)\cos(8\pi)+\cos(-t)\sin(8\pi)]+[\cos(-t)\cos(2\pi)+\sin(-t)\sin(2\pi)]+\frac{\tan(-t)-\tan(5\pi)}{1+\tan(-t)\tan(5\pi)}[/tex]
We are about to use that cos(even*pi) is 1 and sin(even*pi) is 0 so tan(odd*pi)=0:
[tex]-[\sin(-t)(1)+\cos(-t)(0)]+[\cos(-t)(1)+\sin(-t)(0)]+\frac{\tan(-t)-0}{1+\tan(-t)(0)[/tex]
Cleaning up the algebra:
[tex]-[\sin(-t)]+[\cos(-t)]+\frac{\tan(-t)}{1}[/tex]
Cleaning up more algebra:
[tex]-\sin(-t)+\cos(-t)+\tan(-t)[/tex]
Applying that sine and tangent is odd while cosine is even. That is,
sin(-x)=-sin(x) and tan(-x)=-tan(x) while cos(-x)=cos(x):
[tex]\sin(t)+\cos(t)-\tan(t)[/tex]
Making the substitution the problem wanted us to:
[tex]a+b-c[/tex]
Just for fun you could have wrote c as a/b too since tangent=sine/cosine.
A standard deck of 52 cards contains four suits: clubs, spades, hearts, and diamonds. Each deck contains an equal number of cards in each suit. Rochelle chooses a card from the deck, records the suit, and replaces the card. Her results are shown in the table.
How does the experimental probability of choosing a heart compare with the theoretical probability of choosing a heart?
A.The theoretical probability of choosing a heart is 1/16 greater than the experimental probability of choosing a heart.
B.The experimental probability of choosing a heart is 1/16 greater than the theoretical probability of choosing a heart.
C.The theoretical probability of choosing a heart is 1/26 greater than the experimental probability of choosing a heart.
D.The experimental probability of choosing a heart is 1/26 greater than the theoretical probability of choosing a heart.
52 cards / 4 suits = 13 cards of each suit.
Theoretically picking a heart would be 13/52 = 1/4 probability.
Experimentally she picked 15 hearts out of 80 total tries. for a 15/80 = 3/16 probability, which is less than the theoretical probability.
1/4 - 3/16 = 1/16
The answer is A.
Answer:
The right answer is A - The theoretical probability of choosing a heart is StartFraction 1 over 16 EndFraction greater than the experimental probability of choosing a heart.
Suppose that E and F are two events and that Upper P left parenthesis Upper E and Upper F right parenthesis equals 0.1 and Upper P left parenthesis Upper E right parenthesis equals 0.2. What is Upper P left parenthesis F|E right parenthesis
Answer:
The value of [tex]P(\frac{F}{E})[/tex] is [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Given,
[tex]P(E\cap F)=0.1[/tex]
[tex]P(E)=0.2[/tex]
Thus, by the conditional probability formula,
[tex]P(\frac{F}{E})=\frac{P(E\cap F)}{P(E)}[/tex]
By substituting the values,
[tex]P(\frac{F}{E})=\frac{0.1}{0.2}[/tex]
[tex]=\frac{1}{2}[/tex]
A psychologist is interested in constructing a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 64 of the 708 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible. a. With 99% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between and . b. If many groups of 708 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about percent will not contain the true population propo
Final answer:
To construct a 99% confidence interval for the proportion of people who accept the theory, you need to calculate the point estimate, standard error, margin of error, lower bound, and upper bound.
Explanation:
The psychologist wants to construct a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. Let's calculate the confidence interval:
Calculate the point estimate: 64 out of 708 people agreed with the theory, so the estimated proportion is 64/708 = 0.0904.
Calculate the standard error: SE = sqrt((0.0904 * (1 - 0.0904)) / 708) = 0.0091.
Calculate the margin of error: ME = 2.57 * 0.0091 ≈ 0.0234.
Calculate the lower bound: Lower bound = 0.0904 - 0.0234 ≈ 0.067.
Calculate the upper bound: Upper bound = 0.0904 + 0.0234 ≈ 0.113.
So, with 99% confidence, the proportion of all people who accept the theory is between 0.067 and 0.113.
Ms. Smith drove a total of 700 miles on a business to
If her car averaged 35 miles per gallon of gasoline a
gasoline cost $1.25 per gallon, what was the cost in
dollars of the gasoline for the trip?
Answer:
The total amount of gas Ms. Smith used on the trip is $25.
Step-by-step explanation:
To solve this problem, we first must figure out how many gallons of gas Ms. Smith used on her trip. If she drove 700 miles and averaged 35 miles per gallon, if we divide 700 by 35 we can figure out how many gallons she used.
700/35 = 20
Thus, Ms. Smith used 20 gallons on her trip. Next, to figure out what the cost of the gasoline was, we must multiply the number of gallons (20) by the cost of gasoline per gallon ($1.25).
20 gallons * $1.25/gal = $25
Therefore, the cost of gasoline for the trip was $25.
Hope this helps!
Suppose that 50% of all adults regularly consume coffee, 65% regularly consume carbonated soda, and 45% regularly consumes both coffee and soda. (a) What is the chance a randomly selected adult regularly drinks coffee but doesn't drink soda?
Answer: There is a chance of 5% of adult regularly drinks coffee but doesn't drink soda.
Step-by-step explanation:
Since we have given that
Probability of all adults consume coffee P(C) = 50%
Probability of all adults consume carbonated soda P(S) = 65%
Probability of all adults consumes both coffee and soda P(C∩S) = 45%
We need to find the probability that adult regularly drinks coffee but doesn't drink soda.
So, it is talking about difference of sets in which we consider only one set completely i.e. it contains all element of one set but never contains any element of another set.
Here, P( Coffee - Soda) = P(C)-P(C∩S)
[tex]P(C-S)=0.50-0.45=0.05=0.05\times 100\%=5\%[/tex]
Hence, there is a chance of 5% of adult regularly drinks coffee but doesn't drink soda.
The probability that a randomly selected adult regularly drinks coffee but doesn't drink soda, given the provided data, is calculated to be 5%.
Explanation:To find the probability that a randomly selected adult regularly drinks coffee but doesn't drink soda, we start with the information given: 50% of all adults regularly consume coffee, 65% regularly consume carbonated soda, and 45% regularly consume both coffee and soda. To find the probability of adults who consume coffee but not soda, we subtract the percentage of adults who consume both from the percentage of those who consume coffee. This is because those who consume both are also counted in the total number of coffee drinkers.
So, the calculation is as follows:
Percentage of adults who drink coffee: 50%Percentage of adults who drink both coffee and soda: 45%Percentage of adults who drink coffee but not soda: 50% - 45% = 5%Therefore, the chance that a randomly selected adult regularly drinks coffee but doesn't drink soda is 5%.
Stackable polystyrene cups have a height h1=12.5 cm. Two stacked cups have a height of h2=14 cm. Three stacked cups have a height of h3=15.5 cm. Find the equation for hx= 1.5 x+ Your friend is 200 cm tall. Find out how many cups you will need to reach the height of your friend. cups
Answer:
Approximately 59 stacked cups.
Step-by-step explanation:
Given,
Height of a cup = 12.5 cm,
Two stacked cups = 14 cm,
Three stacked cups = 15.5 cm,
........, so on,....
Thus, there is an AP that represents the given situation,
12.5, 14, 15.5,....
First term is, a = 12.5,
Common difference, d = 1.5 cm,
Thus, the height of x cups is,
[tex]h(x) = a+(x-1)d = 12.5 + (x-1)1.5 = 1.5x + 11[/tex]
According to the question,
h(x) = 200
⇒ 1.5x + 11 = 200
⇒ 1.5x = 189
⇒ x = 59.3333333333 ≈ 59,
Hence, approximately 59 stacked cups will need.
Answer:
hx = 1.5cm . x + 11 cm
126 cups
Step-by-step explanation:
We have the following ordered pairs (x, hx).
(1, 12.5 cm)(2, 14 cm)(3, 15.5 cm)We are looking for a linear equation of the form:
hx = a.x + b
where,
a is the slope
b is the y-intercept
To find the slope, we take any pair of ordered values and replace their values in the following expression.
[tex]a=\frac{\Delta hx }{\Delta x} =\frac{h2-h1}{2-1} =\frac{14cm-12.5cm}{2-1} =1.5cm[/tex]
Now, the general form is:
hx = 1.5cm . x + b
We can take any ordered pair and replace it in this expression to find b. Let's use h1.
h1 = 1.5cm . x1 + b
12.5 cm = 1.5 cm . 1 + b
b = 11 cm
The final equation is:
hx = 1.5cm . x + 11 cm
If hx = 200 cm,
200 cm = 1.5cm . x + 11 cm
189 cm = 1.5cm . x
x = 126
The cost of energy delivered to residences by electrical transmission varies from $0.070/kWh to $0.258/kWh throughout the United States; $0.110/kWh is the average value. (a) At this average price, calculate the cost of leaving a 40-W porch light on for two weeks while you are on vacation? (b) At this average price, calculate the cost of making a piece of dark toast in 3.00 min with a 925-W toaster?
Answer:
(a) $1.48
(b) $0.0051 . . . . slightly more than 1/2¢
Step-by-step explanation:
cost = (kW)(hours)(cost/kWh)
(a) cost = (0.040 kW)(24 h/da)(14 da/wk)($0.11/kWh) ≈ $1.478
The cost of leaving the porch light on is about $1.48.
__
(b) cost = (0.925 kW)(3 min)/(60 min/h)($0.11/kWh) ≈ $0.00509
The cost of toasting bread is about 0.5 cents.
The cost of leaving a 40-W porch light on for two weeks at an average electrical cost is $1.4784. The cost of making a toast using a 925-W toaster for 3 minutes at the same price is $0.0050875.
Explanation:(a) We start by converting the wattage of the porch light, which is 40 Watts, or 40 Joules per second, into kilowatts. So, 40 W = 0.04 kW. Now, to find how much energy is used over two weeks (14 days), we need to multiply the wattage in kilowatts by the number of hours in two weeks.
There are 24 hours in a day and 60 minutes in an hour, so there are 14 days * 24 hours/day = 336 hours in two weeks.
Energy used = Power * Time = 0.04 kW * 336 hours = 13.44 kWh.
The cost is then Energy * Cost per unit of energy = 13.44 kWh * $0.110/kWh = $1.4784.
(b) In the case of the toaster, we repeat the same procedure, but using its wattage and the number of minutes it’s used to make toast. Convert the power to kW: 925 W = 0.925 kW. Convert the time to hours: 3 minutes = 0.05 hours.
Energy used = Power * Time = 0.925 kW * 0.05 hours = 0.04625 kWh.
Cost = Energy * Cost per unit of energy = 0.04625 kWh * $0.110/kWh = $0.0050875, or just over half a cent to make a piece of toast.
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Use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in the table. If one order is selected, find the probability of getting an order from Restaurant A or an order that is accurate. Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint events? The probability of getting an order from Restaurant A or an order that is accurate is ...? Round to 3 decimal places.
Restaurant A Restaurant B Restaurant C Restaurant D
Order Accurate 321 276 235 126
Order Not Accurate 32 56 40 11
The probability of getting an order from Restaurant A is [tex]0.321[/tex]
The probability of getting an order accurate in all the Restaurants is [tex]0.873[/tex]
The Probability of getting an order accurate in the Restaurant A is [tex]0.909[/tex]
Selecting an order from Restaurant A and selecting an accurate order are disjoint events
Total number of orders [tex]=321+276+235+126+32+56+40+11=1097[/tex]
The probability of getting an order from Restaurant A (the accurate +the not accurate) is [tex]\dfrac{321+32}{1097}=0.321[/tex]
Total Order Accurate [tex]=321+276+235+126=958[/tex]
Probability of getting an order accurate is [tex]\dfrac{958}{1097}=0.873[/tex]
Probability of getting an order accurate in the Restaurant A is [tex]\dfrac{321}{353}=0.909[/tex]
Learn more about probability.
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To find the probability of getting an order from Restaurant A or an accurate order, sum the values of Restaurant A and the accurate orders and divide it by the total number of orders. The events of selecting an order from Restaurant A and selecting an accurate order are not disjoint.
Explanation:To find the probability of getting an order from Restaurant A or an accurate order, we need to sum the values of Restaurant A and the accurate orders and divide it by the total number of orders.
Probability of getting an order from Restaurant A = (321+32)/(321+276+235+126+32+56+40+11)
Probability of getting an accurate order = (321+276+235+126)/(321+276+235+126+32+56+40+11)
Since both events can occur simultaneously (an order can be from Restaurant A and accurate), they are not disjoint. The final probability is the sum minus the probability of their intersection.
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Suppose you're taking an Honors Algebra 1 multiple choice test. The test consists of 40 questions, each having 5 options. If you guess at all 40 questions, what is the mean of the number of correct answers?
Answer:8
Step-by-step explanation:
We have given test consists of 40 multiple choice questions having five options for each question.
Suppose there is one correct answer to each question
therefore probabilty of getting a correct answer on making a guess is [tex]\frac{1}{5}[/tex]
i.e. 1 out of 5 questions is correct
Using binomial distribution
where n=40 p=[tex]\frac{1}{5}[/tex]
mean of binomial distribution is np
therefore mean of no of correct answers=[tex]40\times \frac{1}{5}[/tex]
=8
The equation of a circle is given below. Identify the radius and center.
x^2 + y^2 - 6x -2y +1 = 0
Answer:
The center is (3,1) and the radius is 3.
Step-by-step explanation:
The goal is to write in [tex](x-h)^2+(y-k)^2=r^2[/tex] because this tells us the center (h,k) and the radius r.
So we are going to need to complete the square 2 times here, once for x and the other time for y.
I'm going to use this formula to help me to complete the square:
[tex]x^2+bx+(\frac{b}{2})^2=(x+\frac{b}{2})^2[/tex].
So first step:
I'm going to group my x's together and my y's.
[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }+1=0[/tex]
Second step:
I'm going to go ahead and subtract that one on both sides. Those blanks are there because I'm going to fill them in with a number so that I can write the x part and y part as a square. Remember whatever you add on one side you must add on the other. So I'm going to put 2 more blanks to fill in on the opposite side of the equation.
[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }=-1+\text{ ___ }+\text{ ___}[/tex]
Third step:
Alright first blank I'm putting (-6/2)^2 due to my completing the square formula. That means this value will also go on the other side in on of those blanks.
In the second blank I'm going to put (-2/2)^2 due to the completing the square formula. This must also go on one of the blanks on the other side.
So we have:
[tex]x^2-6x+(\frac{-6}{2})^2+y^2-2y+(\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]
Fourth step:
Don't make this more hurtful than it already is. Just use the formula drag down the things inside the square. Remember this:
[tex]x^2+bx+(\frac{b}{2})^2[/tex]
equals
[tex](x+\frac{b}{2})^2[/tex].
We are applying that left hand side there (that bottom thing I just wrote).
Let's try it:
[tex](x+\frac{-6}{2})^2+(y+\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]
Fifth step:
The hard part is out of the way.
This is just a bunch of simplifying now:
[tex](x-3)^2+(y-1)^2=-1+9+1[/tex]
[tex](x-3)^2+(y-1)^2=9[/tex]
The center is (3,1) and the radius is 3.
Evaluate the circulation of G⃗ =xyi⃗ +zj⃗ +3yk⃗ around a square of side length 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.
Given that
[tex]\vec G(x,y,z)=xy\,\vec\imath+z\,\vec\jmath+3y\,\vec k[/tex]
has a fairly simple curl,
[tex]\nabla\times\vec G(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]
we can take advantage of Stokes' theorem by transforming the line integral of [tex]\vec G[/tex] along the boundary of the square (call it [tex]S[/tex]) to the integral of [tex]\nabla\times\vec G[/tex] over [tex]S[/tex] itself. Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=u\,\vec\jmath+v\,\vec k[/tex]
with [tex]-\dfrac92\le u\le\dfrac92[/tex] and [tex]-\dfrac92\le v\le\dfrac92[/tex]. Then take the normal vector to [tex]S[/tex] to be
[tex]\vec s_u\times\vec s_v=\vec\imath[/tex]
so that
[tex]\displaystyle\int_{\partial S}\vec G\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec G)\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_{-9/2}^{9/2}\int_{-9/2}^{9/2}(2\,\vec\imath)\cdot(\vec\imath)\,\mathrm du\,\mathrm dv=\boxed{162}[/tex]
We have,the circulation of [tex]G=xyi+zi+3yk[/tex] around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is
[tex]\int_{\theta} G.dr=162[/tex]
From the Question we have the equation to be
[tex]G=xyi+zi+3yk[/tex]
Therefore
[tex]\triangle *G=\begin{Bmatrix}i & j & k\\\frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\xy&z&3y\end{Bmatrix}[/tex]
Where
For i
[tex]i(\frac{d}{dy}*3y-(\frac{d}{dz}*xy))\\\\2i[/tex]
For j
[tex]j(\frac{d}{dx}*3y-(\frac{d}{dz}*xy))[/tex]
[tex]0j[/tex]
For z
[tex]z(\frac{d}{dy}*xy-(\frac{d}{dx}*z))[/tex]
[tex]xk[/tex]
Therefore
[tex]\triangle *G=2i-xk[/tex]
Generally considering [tex]\theta[/tex] as the origin
We apply Stoke's Theorem
[tex]\int_{\theta} G.dr=\int_{\theta}(\triangle *G) i ds[/tex]
[tex]\int_{\theta} G.dr=\int_{\theta}(=2i-xk) i ds[/tex]
[tex]\int_{\theta} G.dr=\int2ds[/tex]
[tex]\int_{\theta} G.dr=2*9^2[/tex]
[tex]\int_{\theta} G.dr=162[/tex]
Therefore
The circulation of [tex]G=xyi+zi+3yk[/tex] around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is
[tex]\int_{\theta} G.dr=162[/tex]
A casino advertises that it gives a 95.2â% payback on slotâ machines, and the balance is retained by the casino. If the amount retained by the casino is â$4820â, find the total amount played on the slot machines.
Answer: The total amount played on the slot machines is $100416.67.
Step-by-step explanation:
Since we have given that
Percentage of amount payback on slot machines = 95.2%
Percentage of amount retained by the casino = 100 - 95.2 =4.8%
Amount retained by the casino = $4820
Let the total amount played on the slot machines be 'x'.
According to question, we get that
[tex]\dfrac{4.8}{100}\times x=\$4820\\\\0.048x=\$4820\\\\x=\dfrac{4820}{0.048}\\\\x=\$100416.67[/tex]
Hence, the total amount played on the slot machines is $100416.67.
Final answer:
To find the total amount played on slot machines with a casino retention of $4820 and a payback percentage of 95.2%, you solve the equation 0.048 * T = $4820 to get T = $100,416.67.
Explanation:
The student asked how to find the total amount played on slot machines if the casino retains $4820, given a payback percentage of 95.2%. To solve this problem, we use the concept of averages and percentages. The amount retained by the casino represents 100% minus the payback percentage, which is 4.8% in this case.
Let's denote the total amount played on the slot machines as T. The equation to find T is:
4.8% of T = $4820
We convert the percentage to a decimal and solve for T:
0.048 * T = $4820
T = $4820 / 0.048
T = $100,416.67
Therefore, the total amount played on the slot machines is $100,416.67
[10] In the following given system, determine a matrix A and vector b so that the system can be represented as a matrix equation in the form AX = b. In the given linear system, solve for y without solving for X, Z and w by using Cramer's rule, x + y + 2 + 2w = 3. -7x – 3y + 5z - 8w = -3 4x + y + z + w = 6 3x + 7y - Z + w = 1
Answer:
[tex]y=-\frac{158}{579}[/tex]
Step-by-step explanation:
To find the matrix A, took all the numeric coefficient of the variables, the first column is for x, the second column for y, the third column for z and the last column for w:
[tex]A=\left[\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right][/tex]
And the vector B is formed with the solution of each equation of the system:[tex]b=\left[\begin{array}{c}3\\-3\\6\\1\end{array}\right][/tex]
To apply the Cramer's rule, take the matrix A and replace the column assigned to the variable that you need to solve with the vector b, in this case, that would be the second column. This new matrix is going to be called [tex]A_{2}[/tex].
[tex]A_{2}=\left[\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right][/tex]
The value of y using Cramer's rule is:
[tex]y=\frac{det(A_{2}) }{det(A)}[/tex]
Find the value of the determinant of each matrix, and divide:
[tex]y==\frac{\left|\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right|}{\left|\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right|} =\frac{158}{-579}[/tex]
[tex]y=-\frac{158}{579}[/tex]
(b) dy/dx = (x - y+ 1)^2
Substitute [tex]v(x)=x-y(x)+1[/tex], so that
[tex]\dfrac{\mathrm dv}{\mathrm dx}=1-\dfrac{\mathrm dy}{\mathrm dx}[/tex]
Then the resulting ODE in [tex]v(x)[/tex] is separable, with
[tex]1-\dfrac{\mathrm dv}{\mathrm dx}=v^2\implies\dfrac{\mathrm dv}{1-v^2}=\mathrm dx[/tex]
On the left, we can split into partial fractions:
[tex]\dfrac12\left(\dfrac1{1-v}+\dfrac1{1+v}\right)\mathrm dv=\mathrm dx[/tex]
Integrating both sides gives
[tex]\dfrac{\ln|1-v|+\ln|1+v|}2=x+C[/tex]
[tex]\dfrac12\ln|1-v^2|=x+C[/tex]
[tex]1-v^2=e^{2x+C}[/tex]
[tex]v=\pm\sqrt{1-Ce^{2x}}[/tex]
Now solve for [tex]y(x)[/tex]:
[tex]x-y+1=\pm\sqrt{1-Ce^{2x}}[/tex]
[tex]\boxed{y=x+1\pm\sqrt{1-Ce^{2x}}}[/tex]
A publisher reports that 55% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 200 found that 46% of the readers owned a particular make of car. Determine the P-value of the test statistic. Round your answer to four decimal places.
The p-value is obtained by measuring how extreme the observed test statistic is compared to what is expected under the null hypothesis. In this case, a p-value less than the significance level (0.05) leads to the rejection of the null hypothesis, indicating that the actual proportion of readers owning the car is different from the reported 55%.
Explanation:Given that the publisher claims that 55% of their readers own a particular make of car, this would be our null hypothesis (H0: p = 0.55), with the actual percentage of their readers owning a car being the alternate hypothesis (Ha: p ≠ 0.55). The sample found that 46% of 200 readers owned the car, so our sample proportion (p') is 0.46.
Using these values, we can calculate the test statistic (Z), which measures the number of standard deviations p' is from p under the null hypothesis. Once we have our test statistic, we can then determine the p-value.
The observed value of the test statistic falls into the critical region. The p-value is the probability that a test statistic will take on a value as extreme (or more extreme) than the observed value of the test statistic calculated from your sample data. If the p-value is less than or equal to the level of significance, we reject the null hypothesis. From the available reference, it is found that the p-value is less than the significance level of 0.05, and therefore, we would reject the null hypothesis and conclude that the proportion is different from 55% based on this sample data.
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The Fratellis will be arrested if and only if the police believe Chunk. Question 9 options: 1) p ↔ q 2) p ∨ q 3) p → q 4) ~p 5) p ∧ q
Answer:
Option 1) p ←→ q
Step-by-step explanation:
The given statement is a combination of two phrases with a connective word.
Phrase 1 - The Fratellis will be arrested.
We consider this phrase as phrase p.
Phrase 2 - If the police believe chunk.
Phrase q.
Now these statements p and q are connected with a connecting word "If only and only".
When two phrases are connected by the connecting words "Only and only if", this connecting word is called as biconditional operator.
p (connecting word) q
p ( If and only if) q
p ←→ q
Therefore, Option 1) p ←→ q is the answer.
UESTION 2 120 MARKS Underground cable breakdown often occur due to the unpredictable deterioration rate of the cable insulation. Cable replacement can be very costly without regular mainte A 9-month study on an underground cable insulation had been conducted to check for the length of the cable insulation that have been deteriorated using fluorescence microscope in order to estimate the maintenance cycle. Table 1 shows the data measured every 3 months during the study Table 1 Time, t (month) Deteriorated cable insulation length, I 1.35 3.72 7.28 15.45 (a) Generate a third order polynomial using Newton's interpolation method to describe the variable of the deteriorated cable insulation length, (t) with respect to time (in month). Write the polynomial in the form of At3BtCt D, where [10 marks] A, B, C and D are constants. (b) According to regulation, the cable insulation requires replacement if it is degraded by 6 mm. Calculate the replacement time using Newton Raphson method with initial guess of 6 months from the polynomial generated in part (a). Perform THREE iterations only and calculate true percent relative error for the final iteration answer [10 marks if the true value of the replacement time is 5 months.
1) Use power series to find the series solution to the differential equation y'+2y = 0 PLEASE SHOW ALL YOUR WORK, OR RISK LOSING ALL POINTS!!
If
[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]
then
[tex]y'=\displaystyle\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n[/tex]
The ODE in terms of these series is
[tex]\displaystyle\sum_{n=0}^\infty(n+1)a_{n+1}x^n+2\sum_{n=0}^\infty a_nx^n=0[/tex]
[tex]\displaystyle\sum_{n=0}^\infty\bigg(a_{n+1}+2a_n\bigg)x^n=0[/tex]
[tex]\implies\begin{cases}a_0=y(0)\\(n+1)a_{n+1}=-2a_n&\text{for }n\ge0\end{cases}[/tex]
We can solve the recurrence exactly by substitution:
[tex]a_{n+1}=-\dfrac2{n+1}a_n=\dfrac{2^2}{(n+1)n}a_{n-1}=-\dfrac{2^3}{(n+1)n(n-1)}a_{n-2}=\cdots=\dfrac{(-2)^{n+1}}{(n+1)!}a_0[/tex]
[tex]\implies a_n=\dfrac{(-2)^n}{n!}a_0[/tex]
So the ODE has solution
[tex]y(x)=\displaystyle a_0\sum_{n=0}^\infty\frac{(-2x)^n}{n!}[/tex]
which you may recognize as the power series of the exponential function. Then
[tex]\boxed{y(x)=a_0e^{-2x}}[/tex]
Solution of differential equation is, [tex]y=e^{-2x}+c[/tex]
Given differential equation is,
[tex]y'+2y=0\\\\\frac{dy}{dx}+2y=0[/tex]
Using separation of variable.
[tex]\frac{dy}{y}=-2dx[/tex]
Integrating both side.
[tex]ln(y)=--2x\\\\y=e^{-2x}+c[/tex]
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In an upcoming race, the top 3 finishers will be recognized with the same award. There are 12 people entered in the race.
How many ways can the top 3 racers be grouped from the 12 people?
There are __[blank]__ ways the top 3 racers can be grouped from the 12 people.
Any help is super appreciated. Thank you in advance
Answer:
220
Step-by-step explanation:
Given,
The total number of people = 12,
Out of which, top 3 finishers will be recognized with the same award,
So, the total possible way = total combination of 3 people out of 12 people
[tex]=^{12}C_3[/tex]
[tex]=\frac{12!}{3!(12-3)!}[/tex]
[tex]=\frac{12\times 11\times 10\times 9!}{3\times 2\times 9!}[/tex]
[tex]=\frac{1320}{6}[/tex]
[tex]=220[/tex]
Hence, there are 220 ways the top 3 racers can be grouped from the 12 people.
Find a particular solution to y" - y' + 9y = 3 sin 3x
Answer:
cos3x
Step-by-step explanation:
y" - y' + 9y = 3 sin 3x
[tex]D^{2}y-Dy+9y=3 sin3x[/tex]
[tex]y=\frac{3 sin 3x}{(D^{2} -D+9}=3 sin 3x[/tex]
here [tex]D^2[/tex] will be replaced by [tex]\alpha^2[/tex] where [tex]\alpha[/tex] is coefficient of x
[tex]y=\frac{3 sin 3x}{-3^{2} -D+9}[/tex]
[tex]y=-3\frac{sin 3x}{D}[/tex]
[tex]y=-3\int\ {sin 3x} \, dx[/tex]
[tex]y=-3\frac{cos3x}{-3}[/tex]
y=cos3x
hence Particular solution is cos3x
1) For A = {a, b, c, d, e} and B = {yellow, orange, blue, green, white, red, black}. a) Define a relation R from A to B that is a function and contains at least 4 ordered pairs. b) What is the domain of this function? c) What is the range of this function?
Answer: The required function is R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .
Step-by-step explanation: We are given the following two sets :
A = {a, b, c, d, e}
and
B = {yellow, orange, blue, green, white, red, black}.
We are to define a relation R from A to B that is a function and contains at least 4 ordered pairs. Also, to find the domain and range of the function.
(a) Let the function R be defined as follows :
R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}.
Since R contains five ordered pairs, so this will fulfill our criterion. Also, since each first element is associated with one and only one second element, so R defines a function.
(b) We know that
the domain of a function is the set of all the first elements in the ordered pairs, so the domain of the function R will be
D = {a, b, c, d, e}.
(c) We know that
the range of a function is the set of all the second elements in the ordered pairs, so the domain of the function R will be
R = {blue, green, white, black}.
Thus, the required function is R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .
Which of the following is a secant of the circle?
A secant is a line that intersects the curve of a circle at two points.
The line in the picture that passes through two points would be Line EF.
Answer:
EF is the secant line
Step-by-step explanation:
A tangent line of a circle is a line that touches the circle at any point on the circumference of the circle.
A secant line is a line that lies inside the circles . secant line crosses the circumference of the circle at two points. The secant line has not end points.
In the given diagram, secant line is EF because it has no end points and it intersects the circle at two points
EF is the secant line
Emily was going to sell all of her stamp collection to buy a video game. After selling half of them she changed her mind. She then bought seventeen more. Write an expression for how many she has now.
i think you have to write an expression to this...
Answer:
s/2 +17
Step-by-step explanation:
If s represents the number of stamps Emily started with, then the number she had after selling half of them is ...
s/2
After purchasing 17 more, she had ...
s/2 +17
could someone please help and explain .
Answer:
WXYZ = Ro(180°, (2, -3))(ABCD)
Step-by-step explanation:
A reflection across two perpendicular lines (y=-3, x=2) is equivalent to reflection across their point of intersection. That, in turn, is equivalent to rotation 180° about that point of intersection.
Your double reflection is equivalent to rotation 180° about (2, -3).
Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) to three decimal places using a graphing utility and compare the results. f(x) = x5 + x − 6
Answer:
There is only one real zero and it is located at x = 1.359
Step-by-step explanation:
After the 4th iteration the solution was repeating the first 3 decimal places. The formula for Newton's Method is
[tex]x_{n}-\frac{f(x_{n}) }{f'(x_{n}) }[/tex]
If our function is
[tex]f(x)=x^5+x-6[/tex]
then the first derivative is
[tex]f'(x)=5x^4+1[/tex]
I graphed this on my calculator to see where the zero(s) looked like they might be, and saw there was only one real one, somewhere between 1 and 2. I started with my first guess being x = 1.
When I plugged in a 1 for x, I got a zero of 5/3.
Plugging in 5/3 and completing the process again gave me 997/687
Plugging in 997/687 and completing the process again gave me 1.36976
Plugging in 1.36976 and completing the process again gave me 1.359454
Plugging in 1.359454 and completing the process again gave me 1.359304
Since we are looking for accuracy to 3 decimal places, there was no need to go further.
Checking the zeros on the calculator graphing program gave me a zero of 1.3593041 which is exactly the same as my 5th iteration!
Newton's Method is absolutely amazing!!!
Using Newton's Method with an initial guess of 1.5, approximations for f(x) = 0 are x ≈ 1.189, close to the actual zeros.
To approximate the zero(s) of the function f(x) = x^5 + x - 6 using Newton's Method, we start with an initial guess, x0. The formula for the iterative step is:
x_(n+1) = x_n - f(x_n) / f'(x_n),
where f'(x) is the derivative of f(x). In this case, f'(x) = 5x^4 + 1.
1. Choose an initial guess, say x0 = 1.5.
2. Iterate using the formula: x_(n+1) = x_n - (x_n^5 + x_n - 6) / (5x_n^4 + 1).
Continue these iterations until two successive approximations differ by less than 0.001. It may take several iterations to reach this level of accuracy.
After reaching a sufficiently accurate result, use a graphing utility to confirm the zero(s) to three decimal places. This will help ensure the accuracy of the approximation from Newton's Method. The actual zeros of the function are approximately x ≈ 1.189, x ≈ -1.187, and x ≈ 1.999. Compare the results from Newton's Method to these values.
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In the year 1985, a house was valued at $113,000. By the year 2005, the value had appreciated exponentially to $155,000. What was the annual growth rate between 1985 and 2005 (Rourid your answer to two decimal places.) Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010? (Round your answer to the nearest dolar.)
Answer:
The growth rate is 1.02; the value of the house in 2010 is $185,388
Step-by-step explanation:
This is an exponential growth equation, therefore, it follow the standard form:
[tex]y=a(b)^x[/tex]
where y is the value of the house after a certain number of years,
a is the initial value of the house,
b is the growth rate, and
x is the year number.
We are going to make this easy on ourselves and call year 1985 year 0. Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25. We will make these the x coordinates in our coordinate pairs.
(0, 113000) and (20, 155000)
Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:
[tex]113000=a(b)^0[/tex]
Anything raised to the 0 power is equal to 1, so
113000 = a(1) and
a = 113000
Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:
[tex]155000=113000(b)^{20}[/tex]
Start by dividing both sides by 113000 to get a decimal:
[tex]1.371681416=b^{20}[/tex]
To solve for b, we have to undo that power of 20 by taking the 20th root of b. Because this is an equation, we have to take the 20th root of both sides:
[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]
The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:
b = 1.0159 which rounds to
b = 1.02 This is our growth rate.
Now we can use this growth rate and the value of a we found to write the model for our situation:
[tex]y=113000(1.02)^x[/tex]
If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:
[tex]y=113000(1.02)^{25}[/tex]
Raise 1.02 to the 25th power and get:
y = 113000(1.640605994) and multiply to get a final value of
y = $185,388
The annual exponential growth rate from 1985 to 2005 is 1.40%. Using this rate, the estimated value of the house in 2010 would be approximately $176,927.
Explanation:In order to calculate the exponential growth rate, we use the formula: R = (final value/initial value)^(1/n) - 1, where R is the annual rate, n is the number of years. So, R = (155,000/113,000)^(1/20) - 1 = 0.0140 or 1.40%.
To find the value of the house in 2010, we use the exponential growth formula: future value = present value * (1 + annual rate)^n. The value in 2010 would be $155,000 * (1 + 0.0140)^5 = $176,927 (rounded to the nearest dollar).
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6 Points possible: 3. Total attempts: 5
For the data shown, answer the questions. Round to 2 decimal places.
x
7.3
11.7
21.7
18.8
23.2
20.7
29.7
21.2
10.6
Find the mean:
Find the median:
Find the standard deviation:
Answer:
Mean=18.32
Median=20.7
Standard deviation=7.10
Step-by-step explanation:
Mean is the average of the sample size. It is calculated by dividing the sum of all the observations by the number of observations.
[tex]mean=\frac{7.3+11.7+21.7+18.8+23.2+20.7+21.2+10.6}{9}[/tex]
[tex]\Rightarrow mean=18.32[/tex]
Median is the middle value of the observations if the number of observations is odd. If the number of the observations is even then it is the middle value and the next observations average. The values need to be arranged in ascending order.
Therefore the observations become
[tex]7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7[/tex]
In this case the number of observations is 9 which is odd
Therefore, the median is 20.7 i.e., the fifth observation
[tex]Standard\ deviation=\sqrt \frac{\sum_{i=1}^{N}\left ( x_{i}-\bar{x} \right )^2}{N-1}[/tex]
[tex]where,\ N=Number\ of\ observations\\\bar x=mean\\x_{i}=x_{1}+x_{2}+x_{3}+x_{4}.........x_{N}[/tex]
[tex]\left ({x_{i}}-\bar {x}\right )}^2[/tex]
[tex]\\121.49\\43.85\\11.41\\0.23\\23.79\\5.65\\129.45\\8.28\\59.63\\\sum_{i=1}^{N}\left ({x_{i}}-\bar {x}\right )}^2=403.80\\N-1=9-1\\=8\\\therefore Standard\ deviation=\sqrt {\frac{403.80}{8}}=7.10[/tex]
Standard deviation=7.10
Final answer:
This detailed answer addresses how to find the mean, median, and standard deviation for a given data set in a High School Mathematics context.
Explanation:
Mean: To find the mean, add up all the values and then divide by the total number of values. Add all the values provided: 7.3 + 11.7 + 21.7 + 18.8 + 23.2 + 20.7 + 29.7 + 21.2 + 10.6 = 164.9. Then, divide by 9 (total values) to get a mean of 164.9/9 = 18.32.
Median: To find the median, arrange the data in numerical order and find the middle value. The data in order is: 7.3, 10.6, 11.7, 18.8, 20.7, 21.2, 21.7, 23.2, 29.7. The median is the middle value, which is 20.7.
Standard Deviation: To calculate the standard deviation, you need to find the variance first. Then, take the square root of the variance. Using the given data above, the standard deviation is approximately 6.94.