Answer:
a) 10912 Yards
b) 9977.93 meters
Step-by-step explanation:
We are given the following information in the question.
1 Meter = 100 Centimeter
1 Inch = 2.54 Centimeter
1 Foot = 12 Inch
1 Yard = 3 Foot
1 Mile = 1760 Yards
The length of race is 6.2 miles.
a) We have to find length of race in yards.
Since we know 1 Mile = 1760 Yards
Thus, 6.2 Miles = 6.2 × 1760 Yards = 10912 Yards
Thus , the race is 10912 yards in length.
b) We have to calculate the length of the race in meters.
Since we know 1 Mile = 1760 Yards
Thus. 6.2 miles = [tex]\frac{(6.2)(1760)(3)(12)(2.54)}{100}[/tex] = 9977.93 meters
Please help I can’t find the primes.
Answer:
the primes are all the even numbers in the equation
Step-by-step explanation:
After a statistics class, Dr. Johnson asked all 51 of her students the degree to which they felt statistics were going to be useful in their eventual career. She had all of her students respond to the following question, "Statistics will be very useful in my future profession," using the following scale: 1 = not at all useful and 7 = very useful. She found that the median response was a 5.5. Given that the median of 5.5 was based on all of her students, the 5.5 would be called __________.
Answer:
The 5.5 would be called a parameter.
Step-by-step explanation:
A parameter in statistics is a data/number/quantity that gives you information about an entire population. Given that Dr. Johnson asked ALL of her students to respond the question and the median of 5.5 was based on ALL of her students, we can say that this number would be a parameter for the population (in this case, Dr. Johnson's class)
In high school, some students have been confused to believe that 22/7 is already the actual value of π or an acceptable approximation. Show that 355/113 is a better approximation in terms of absolute and relative errors.
Answer:
The absolute and relative error of 355/113 compared with π is less than when π is compared with 22/7, that's why 355/113 is a better approximation for the actual value of π.
Step-by-step explanation:
The absolute error is the difference between a value measured and the real value.
abs = π - approximation of π
The relative error indicates how large the absolute error is when compared with the actual value of π.
Now, let's calculate the absolute an relative error for each approximation of π, for simplicity the calculations will be rounded to 4 decimal digits.
rel = abs / π
For 22/7abs = π - 22/7
abs = -0.0013
rel = (π - 22/7) / π
rel = -0.0402 %
For 355/113abs = π - 355/113
abs = -2.6676 x10-7
rel = (π - 355/113) / π
rel = -8.4914 x10-6 %
You can see that both the value of the absolute and relative error for the 355/113 approximation are smaller numbers, in conclusion, 355/113 is a better approximation for π.
A good-quality measuring tape can be off by 0.49 cm over a distance of 23 m. What is the percent uncertainty?33% Part (a) If 44 ± 3 beats are counted (in exactly 30 s) what is the percent uncertainty in the measurement of their heartbeats per minute?
Step-by-step explanation:
First:
(Error/Total )x 100%
(0.49/23) x100%= 2.13%
Second:
(Absolute uncertainty/mean) x100%
(3/44)x100%= 6.81%
To calculate percent uncertainty, divide the uncertainty by the measured value and multiply by 100%. The percent uncertainty for the measuring tape is 0.0213%. The percent uncertainty for heartbeats (converted to per minute) is 6.82%.
Explanation:The subject of this question is determining the percent uncertainty of measurements. Percent uncertainty is calculated by dividing the uncertainty by the measurement and multiplying by 100%.
1. For the measuring tape, the uncertainty is 0.49 cm which first needs to be converted to meters (0.0049 m) since the measurement is given in meters. The percent uncertainty of the measuring tape is therefore (0.0049 m / 23m) * 100% = 0.0213%, a small percentage reflecting a high level of accuracy.
2. For counting heartbeats, the uncertainty is 3 beats and the measurement is 44 beats. Since it's measured in 30 seconds, we should double it to get beats per minute. Therefore, the uncertainty becomes 6 beats and the measurement becomes 88 beats. The percent uncertainty is therefore (6 beats / 88 beats) * 100% = 6.82%, a more considerable uncertainty.
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Find #(A u B u C) if #A = #B = #C = 17, #(A n B) = 5, #(B n C) = 6, #(An C) = 7, and #(A n B n C-2. You may find it helpful to draw a Venn diagram.
Answer:
35
Step-by-step explanation:
Given that A,B, C are three non empty sets.
[tex]n(A) =n(B) =n(C) =17\\n(A \bigcap C) = 7\\n(A \bigcap B) = 5\\n(B \bigcap C) = 6\\n(A \bigcap B \bigcap C) = 7[/tex]
Use the addition theory for finding no of elements in union of two or more sets
We have addition theorem as
[tex]n(AUBUC) = n(A)+n(B)+n(C)-n(A \bigcap B)-n(B \bigcap C)-n(A \bigcap C)+n(A \bigcap B \bigcap C)[/tex]
Now substitute for each entry from the given information
[tex]n(AUBUC) = 17+17+17-5-6-7+2\\= 53-18\\=35[/tex]
Multi step equation and check 12x+3-1=35
Answer:
11/4
Step-by-step explanation:
Step 1: Simplify both sides of the equation.
12x+3−1=35
12x+3+−1=35
(12x)+(3+−1)=35(Combine Like Terms)
12x+2=35
12x+2=35
Step 2: Subtract 2 from both sides.
12x+2−2=35−2
12x=33
Step 3: Divide both sides by 12.
12x / 12 = 33 / 12x =
11 / 4
how many distinguishable a arrangements are there of the letters in "REPRESENTATION"?
Answer: There are 1,816,214,400 ways for arrangements.
Step-by-step explanation:
Since we have given that
"REPRESENTATION"
Here, number of letters = 14
There are 2 R's, 3 E's, 2 T's, 2 N's
So, number of permutations would be
[tex]\dfrac{14!}{2!\times 3!\times 2!\times 2!}\\\\=1,816,214,400[/tex]
Hence, there are 1,816,214,400 ways for arrangements.
A company's sales decreased by 8% this year, to $9015. What were their sales last year? (Round your answer to the nearest penny.)
My answer was 9,736.2, but the correct answer was 9798.91. Could someone explain how did they get to 9798.91?
Answer:
There are two unknow values, your 8% which is the decrease and the main value which is "the sales last year".
It is correct, the number you gave as an answer $9798,91. Let's get the explanation by a rule of three.
Step-by-step explanation:
The 100 % was "the sales last year", the 8% are what the sales decreased but you don't have that number, you have the result in the substraction, $9015. So this are the step by step, hope you understand!
20 units Humulin R insulin in 100 mL of normal saline (NS) to infuse for 20 hours. (Round to the nearest tenth if applicable) a. How many units per hour will be infused? ________ b. How many milliliters per hour will be infused? ________
Answer:
a) 1 unit per hour will be infused.
b) 5mL per hour will be infused.
Step-by-step explanation:
Each question can be solved as a rule of three with direct measures, that means we have a cross multiplication.
a. How many units per hour will be infused?
20 units are going to be infused in 20 hours. How many units are going to be infused each hour?
1 hour - x units
20 hours - 20 units
[tex]20x = 20[/tex]
[tex]x = \frac{20}{20}[/tex]
[tex]x = 1[/tex]
1 unit per hour will be infused.
b. How many milliliters per hour will be infused?
100mL are going to be infused in 20 hours. How many mL are going to be infused each hour?
1 hour - x mL
20 hours - 100 mL
[tex]20x = 100[/tex]
[tex]x = \frac{100}{20}[/tex]
[tex]x = 5[/tex]
5mL per hour will be infused.
The owner of a bike shop that produces custom built bike frames has determined that the demand equation for bike frames is given by the equation
D(q) = –6.10q^2 –5q + 1000
where D(q) is the price in dollars and q is the number of bike frames demanded per week. The supply equation for bike frames is
S(q) = 3.20q^2 + 10q – 80
where q is the quantity the supplier will make available per week in the market when the price is p dollars. Find the equilibrium point (q, p) rounded to the nearest hundredth.
Answer:
equilibrium point (10,340)
Step-by-step explanation:
To find the equilibrium point, equal the demand and the supply:
[tex]D(q)=S(q)\\\\-6.10q^2-5q+1000=3.2q^2+10q-80[/tex]
Reorganize the terms in one side and reduce similar terms:
[tex]3.2q^2+6.1q^2+5q+10q-80-1000=0\\\\9.3q^2+15q-1080=0[/tex]
that's a cuadratic equation, solve with the general formula when:
a=9.3, b=15, c=-1080
[tex]q_{1}=\frac{-b+\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{1}=\frac{-15+\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{1}=\frac{-15+201}{18.6}\\\\q_{1}=\frac{186}{18.6}\\\\q_1=10[/tex]
q can't be negative because it is the quantity of bike frames, so:
[tex]q_{2}=\frac{-b-\sqrt{b^{2}-4ac} }{2a}\\\\q_{2}=\frac{-15-\sqrt{(-15)^{2}-4(9.3)(-1080)} }{2(9.3)}\\\\q_{2}=\frac{-15-201}{18.6}\\\\q_{2}=\frac{-216}{18.6}\\\\[/tex]
This value of q can't be considered.
Then substitute the value of q in D(q) to find the price p:
[tex]D(10) = -6.10(10)^2-5(10) + 1000\\\\D(10)=340=p[/tex]
The equilibrium point (q,p) is (10,340).
To find the equilibrium point in supply and demand equations, calculate where the two equations intersect to determine the equilibrium quantity and price.
Equilibrium Point Calculation:
Set D(q) = S(q) to find the equilibrium point.Replace D(q) and S(q) with their respective equations.Solve for q to find the equilibrium quantity and then substitute q back into either D(q) or S(q) to find the equilibrium price.In words, the scalar product of two vectors can be thought of as the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a. It is used to calculate the product of vector quantities when only the parallel components of each vector contribute (e.g., Work = Force • Displacement). Let ~a = h9, 6.75, 0i and ~b = h2.97, 6.075, 0i. Calculate ~a • ~b.
Answer:
67.73625
Step-by-step explanation:
The dot (scalar) product is also the sum of products of corresponding vector components.
~a • ~b = 9·2.97 +6.75·6.075 +0·0 = 27.73 +41.00625 = 67.73625
The scalar product or dot product of two vectors, ~a and ~b, in Cartesian form is calculated by multiplying the matching components of the two vectors and then adding them. Performing these steps will give us 67.73625, which represents the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.
Explanation:In physics, the scalar product, also known as dot product, of two vectors like ~a = h9, 6.75, 0i and ~b = h2.97, 6.075, 0i can be determined using their magnitudes and the cosine of the angle between them. However, in the given question, the vectors are in Cartesian form (i,j,k coordinate system), and we can calculate their dot product directly. The dot product is calculated by multiplying the respective i, j, and k components of the two vectors and then adding them. Let's do this step by step:
The i component of our dot product is 9*2.97 = 26.73.The j component of our dot product is 6.75*6.075 = 41.00625The k component of our dot product becomes 0*0 = 0 as the k component is non-existent for both vectorsSo, the scalar product ~a • ~b = 26.73 + 41.00625 + 0 = 67.73625
This scalar product gives the product of the magnitude of ~a with the magnitude of the projection of ~b onto the direction of ~a.
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Show that if A and B are similar nxn matrices, then det(A)=det(B).
Step-by-step explanation:
To prove it we just use the definition of similar matrices and properties of determinants:
If [tex] A,B[/tex] are similar matrices, then there is an invertible matrix [tex]C[/tex], such that [tex] A=C^{-1}BC}[/tex] (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:
[tex]det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)[/tex]
[tex]=\frac{1}{det(C)}det(B)det(C)=det(B)[/tex]
(Determinant of a product of matrices is the product of their determinants, and the determinant of [tex]C^{-1}[/tex] is just [tex]\frac{1}{det(C)}[/tex])
(CO 3) On average, the parts from a supplier have a mean of 97.5 inches and a standard deviation of 6.1 inches. Find the probability that a randomly selected part from this supplier will have a value between 87.5 and 107.5 inches. Is this consistent with the Empirical Rule of 68%-95%-99.7%? Probability is 0.90, which is consistent with the Empirical Rule Probability is 0.95, which is inconsistent with the Empirical Rule Probability is 0.90, which is inconsistent with the Empirical Rule Probability is 0.05, which is consistent with the Empirical Rule Flag this Question
Answer:
C. Probability is 0.90, which is inconsistent with the Empirical Rule.
Step-by-step explanation:
We have been given that on average, the parts from a supplier have a mean of 97.5 inches and a standard deviation of 6.1 inches.
First of all, we will find z-score corresponding to 87.5 and 107.5 respectively as:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{87.5-97.5}{6.1}[/tex]
[tex]z=\frac{-10}{6.1}[/tex]
[tex]z=-1.6393[/tex]
[tex]z\approx-1.64[/tex]
[tex]z=\frac{x-\mu}{\sigma}[/tex]
[tex]z=\frac{107.5-97.5}{6.1}[/tex]
[tex]z=\frac{10}{6.1}[/tex]
[tex]z=1.6393[/tex]
[tex]z\approx 1.64[/tex]
Now, we need to find the probability [tex]P(-1.64<z<1.64)[/tex].
Using property [tex]P(a<z<b)=P(z<b)-P(z<a)[/tex], we will get:
[tex]P(-1.64<z<1.64)=P(z<1.64)-P(z<-1.64)[/tex]
From normal distribution table, we will get:
[tex]P(-1.64<z<1.64)=0.94950-0.05050 [/tex]
[tex]P(-1.64<z<1.64)=0.899[/tex]
[tex]P(-1.64<z<1.64)\approx 0.90[/tex]
Since the probability is 0.90, which is inconsistent with the Empirical Rule, therefore, option C is the correct choice.
(Discrete Mathematics) If m and n are nonzero integers, show that (2m+3n)/5mn is a rational number.
Answer:
[tex]\frac{(2m+3n)}{5mn}=\frac{2}{5n}+\frac{3}{5m}[/tex] is a rational number for any m and n; nonzero integers.
Step-by-step explanation:
We have been given that 'm' and 'n' are nonzero integers. We are asked to show that [tex]\frac{(2m+3n)}{5mn}[/tex] is a rational number.
We can rewrite our given number as:
[tex]\frac{2m}{5mn}+\frac{3n}{5mn}[/tex]
Cancelling out common terms:
[tex]\frac{2}{5n}+\frac{3}{5m}[/tex]
Since 'm' and 'n' are nonzero integers, so each part will be a rational number.
We know that sum of two rational numbers is always rational, therefore, our given number is a rational number.
Your friend borrows $100 from you and promises to pay you back $109 in 8 months. What annual percentage rate (A.P.R.) are you charging your friend? Round to the nearest tenth of a percent and write the answer as a decimal
Answer:
[tex]12.0\%=0.12[/tex]
Step-by-step explanation:
We have been given that your friend borrows $100 from you and promises to pay you back $109 in 8 months.
We will use simple interest formula to solve our given problem.
[tex]A=P(1+rt)[/tex], where,
A = Amount after t years,
P = Principal amount,
r = Annual interest rate in decimal form,
t = Time in years.
Convert 8 months to year:
[tex]\frac{8}{12}\text{ year}=\frac{2}{3}\text{ year}[/tex]
[tex]108=100(1+r*\frac{2}{3})[/tex]
[tex]108=100+r*\frac{2}{3}\times 100[/tex]
[tex]108-100+r*\frac{200}{3}[/tex]
[tex]108-100=100-100+r*\frac{200}{3}[/tex]
[tex]8=r*\frac{200}{3}[/tex]
[tex]8\times \frac{3}{200}=r*\frac{200}{3}\times \frac{3}{200}[/tex]
[tex]\frac{24}{200}=r[/tex]
[tex]r=\frac{24}{200}[/tex]
[tex]r=0.12[/tex]
Convert to percent:
[tex]0.12\times 100\%=12\%[/tex]
Therefore, you are charging 12% APR to you friend.
Solve the linear system by Gaussian-Jordan elimination. a) x1 + x2 + 2x3 = 8 b) x – y + 2z - w = -1 – x1 – 2x2 + 3x3 = 1 2x + y - 22 – 2w = -2 3x1 – 7x2 + 4x3 = 10 -x + 2y – 4z +w=1 3x – 3w = -3
Answer:
Solution for the linear system:
a) [tex]X_1=3, X_2=1, X_3=2[/tex]
b) [tex]x= w-1\\y=2z[/tex]
z and w are free, meaning that can have any value, for this reason, this system has infinite solutions.
Step-by-step explanation:
Gaussian-Jordan elimination consists of taking an augmented matrix, and transform it into its Row echelon form by means of row operation. For notation, R_i will be the transform column, and r_i the actual one.
Linear System a)
First, you have to convert the system into matrix notation, in this case, column 1 corresponds to variable x_1, column 2 to x_2, column 3 to x_3 and column 4 to the system constants:
[tex]\left[\begin{array}{cccc}1&1&2&8\\-1&-2&3&1\\3&-7&4&10\end{array}\right][/tex]
Transform to 0 every number in the rows under the first row.Operations:
[tex]R_2=r_1+r_2\\R_3=-3r_1+r_3[/tex]
[tex]\left[\begin{array}{cccc}1&1&2&8\\0&-1&5&9\\0&-10&-2&-14\end{array}\right][/tex]
Transform the -1 of the second row into 1Operations:
[tex]R_2=-r_2[/tex]
[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&-10&-2&-14\end{array}\right][/tex]
Transform to 0 the numbers under the 1 in the second rowOperations:
[tex]R_3=10r_2+r_3[/tex]
[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&-52&-104\end{array}\right][/tex]
Transform the -52 in the row 3 into 1Operations:
[tex]R_3=-\frac{1}{52}r_3[/tex]
[tex]\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&1&2\end{array}\right][/tex]
Now write the matrix like linear equations[tex]x_1+x_2+2x_3=8\\x_2-5x_3=-9\\x_3=2[/tex]
Now that you know the valuo of x_3, you can solve the system starting from the bottom up, then find x_2 and finally x_1[tex]x_3=2\\x_2=-9+5*2=1\\x_1=8-1-4=3[/tex]
Linear System b)
For this system, the process is the same as the above.
Convert the system into matrix form
[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\2&1&-2&-2&-2\\-1&2&-4&1&1\\3&0&0&-3&-3\end{array}\right][/tex]
Operations:
[tex]R_2=-2r_1+r_2\\R_3=r_1+r_3\\R_4=-3r_1+r_4[/tex]
[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&3&-6&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right][/tex]
Operations:
[tex]R_2=\frac{1}{3}r_2[/tex]
[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&1&-2&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right][/tex]
Operations:
[tex]R_3=-r_2+r_3\\R_4=-3r_2+r_4[/tex]
[tex]\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&1&-2&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}\right][/tex]
Now you can write the system as equations:
[tex]x-y+2z-w=-1\\y-2z=0[/tex]
For w and z there is no unique answer, so the system result is expressed in terms of those variables. This system has infinite solutions.
Solution:
[tex]x= w-1\\y=2z[/tex]
z and w are free values.
Suppose you go shopping for a new futon bed for your room. The model you really like happens to be on sale for $400. It's original price is $700. What percent of the original price will you save if you purchase it?
Answer:
If you purchase it, you are going to save 0.43 = 43% of the original price
Step-by-step explanation:
This problem can be solved by a rule of three.
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.
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 problem, we have the following measures:
-The prices
-The percentage that the price represents.
As the value of one measure increases, so do the value of the other. It means that we have a direct rule of three.
The problem states that the model you really like happens to be on sale for $400. It's original price is $700.
You saved $700-$400 = $300.
What percent of the original price will you save if you purchase it?
How much is $300 of $700.
$700 - 1
$300 - x
700x = 300
[tex]x = \frac{300}{700}[/tex]
[tex]x = 0.43[/tex]
If you purchase it, you are going to save 0.43 = 43% of the original price
The population of a suburb grows at a rate proportional to the population. Suppose the population doubles in size from 3000 to 6000 in a 6- month period and continues at the current rate of growth. Find the particular solution to the differential equation with the initial condition P(0)=3000.
The particular solution to the differential equation with the initial condition P(0) = 3000 is given by P = 3000e^((ln(2)/6)t), where P represents the population at time t. The growth rate is proportional to the population, and the constant of proportionality is ln(2)/6. This equation can be derived by solving the differential equation and using the initial condition.
Explanation:In this problem, we are given that the population of a suburb grows at a rate proportional to the population. We are also given that the population doubles in size from 3000 to 6000 in a 6-month period. We need to find the particular solution to the differential equation with the initial condition P(0)=3000.
Let's denote the population at time t as P(t). Since the growth rate is proportional to the population, we can write the differential equation as dP/dt = kP, where k is the proportionality constant.
Integrating both sides of the equation, we get: ∫dP/P = ∫kdt. This gives us ln|P| = kt + C, where C is the constant of integration.
Using the initial condition P(0) = 3000, we can substitute t = 0 and P = 3000 into the equation to get: ln|3000| = 0 + C. Solving for C, we find C = ln|3000|.
Substituting C = ln|3000| back into the equation, we have ln|P| = kt + ln|3000|. Simplifying, we get ln|P| - ln|3000| = kt.
Since ln|P| - ln|3000| = ln|(P/3000)|, we can write the equation as ln|(P/3000)| = kt.
Taking the exponential of both sides, we get |(P/3000)| = e^(kt).
Since the population cannot be negative, we can remove the absolute value sign and write the equation as (P/3000) = e^(kt).
Substituting the doubling in size from 3000 to 6000 in a 6-month period, we have (6000/3000) = e^(k(6)).
Simplifying, we get 2 = e^(6k).
Taking the natural logarithm of both sides, we have ln(2) = ln(e^(6k)).
Using the property ln(a^b) = bln(a), we can rewrite the equation as ln(2) = 6kln(e).
Since ln(e) = 1, we have ln(2) = 6k.
Solving for k, we get k = ln(2)/6.
Substituting k = ln(2)/6 back into the equation, we have (P/3000) = e^((ln(2)/6)t).
Multiplying both sides by 3000, we get P = 3000e^((ln(2)/6)t).
This is the particular solution to the differential equation with the initial condition P(0) = 3000.
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Solve the initial value problem: y'(x)=(4y(x)+25)^(1/2) ,y(1)=6. you can't really tell, but the '1/2' is the exponent
Answer:
[tex]y(x)=x^2+5x[/tex]
Step-by-step explanation:
Given: [tex]y'=\sqrt{4y+25}[/tex]
Initial value: y(1)=6
Let [tex]y'=\dfrac{dy}{dx}[/tex]
[tex]\dfrac{dy}{dx}=\sqrt{4y+25}[/tex]
Variable separable
[tex]\dfrac{dy}{\sqrt{4y+25}}=dx[/tex]
Integrate both sides
[tex]\int \dfrac{dy}{\sqrt{4y+25}}=\int dx[/tex]
[tex]\sqrt{4y+25}=2x+C[/tex]
Initial condition, y(1)=6
[tex]\sqrt{4\cdot 6+25}=2\cdot 1+C[/tex]
[tex]C=5[/tex]
Put C into equation
Solution:
[tex]\sqrt{4y+25}=2x+5[/tex]
or
[tex]4y+25=(2x+5)^2[/tex]
[tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex]
[tex]y(x)=x^2+5x[/tex]
Hence, The solution is [tex]y(x)=\dfrac{1}{4}(2x+5)^2-\dfrac{25}{4}[/tex] or [tex]y(x)=x^2+5x[/tex]
Solve these linear equations in the form y=yn+yp with yn=y(0)e^at.
a. y'-4y=-8
b. y'+4y=8
which one has a steady state?
Answer:
a) [tex]y(t) = y_{0}e^{4t} + 2[/tex]. It does not have a steady state
b) [tex]y(t) = y_{0}e^{-4t} + 2[/tex]. It has a steady state.
Step-by-step explanation:
a) [tex]y' -4y = -8[/tex]
The first step is finding [tex]y_{n}(t)[/tex]. So:
[tex]y' - 4y = 0[/tex]
We have to find the eigenvalues of this differential equation, which are the roots of this equation:
[tex]r - 4 = 0[/tex]
[tex]r = 4[/tex]
So:
[tex]y_{n}(t) = y_{0}e^{4t}[/tex]
Since this differential equation has a positive eigenvalue, it does not have a steady state.
Now as for the particular solution.
Since the differential equation is equaled to a constant, the particular solution is going to have the following format:
[tex]y_{p}(t) = C[/tex]
So
[tex](y_{p})' -4(y_{p}) = -8[/tex]
[tex](C)' - 4C = -8[/tex]
C is a constant, so (C)' = 0.
[tex]-4C = -8[/tex]
[tex]4C = 8[/tex]
[tex]C = 2[/tex]
The solution in the form is
[tex]y(t) = y_{n}(t) + y_{p}(t)[/tex]
[tex]y(t) = y_{0}e^{4t} + 2[/tex]
b) [tex]y' +4y = 8[/tex]
The first step is finding [tex]y_{n}(t)[/tex]. So:
[tex]y' + 4y = 0[/tex]
We have to find the eigenvalues of this differential equation, which are the roots of this equation:
[tex]r + 4 = [/tex]
[tex]r = -4[/tex]
So:
[tex]y_{n}(t) = y_{0}e^{-4t}[/tex]
Since this differential equation does not have a positive eigenvalue, it has a steady state.
Now as for the particular solution.
Since the differential equation is equaled to a constant, the particular solution is going to have the following format:
[tex]y_{p}(t) = C[/tex]
So
[tex](y_{p})' +4(y_{p}) = 8[/tex]
[tex](C)' + 4C = 8[/tex]
C is a constant, so (C)' = 0.
[tex]4C = 8[/tex]
[tex]C = 2[/tex]
The solution in the form is
[tex]y(t) = y_{n}(t) + y_{p}(t)[/tex]
[tex]y(t) = y_{0}e^{-4t} + 2[/tex]
Write a meaningful negation for the following statement.
The shirt I’m wearing to my interview is orange.
Answer:
The negation will be : The shirt I am wearing to my interview is not orange.
Step-by-step explanation:
The negation of the statement means adding not, or nor to the statement.
The given statement is :
The shirt I’m wearing to my interview is orange.
The negation will be : The shirt I am wearing to my interview is not orange.
Is the following statement true of false? Givejustification if
true or a counterexample if false.
Any two quadrilaterals are similar if
theircorresponding angles are congruent.
Answer:
No , any two quadrilaterals may not be similar if their corresponding angles are congruent.
Step-by-step explanation:
We need to check that whether two quadrilaterals are similar if
their corresponding angles are congruent.
A quadrilateral is a polygon having two sides .
Two figures are said to be similar if they have same shape .
Two angles are said to be congruent if they have same measure .
Consider two quadrilaterals : rectangle and square
Each angle of square and rectangle is equal to [tex]90^{\circ}[/tex] . So, their corresponding angles are congruent .
But square and rectangle are not similar as they have different shape .
In Mary's first math test she scored 87%. The mean and standard deviation for the class were 71% and 18% respectively. In her second math test, Mary scored 66%. The mean and standard deviation for the class were 53% and 14% respectively. In which test did Mary do better relative to the rest of the class? Explain your reasoning. (Hint: find the z-scores corresponding to her two test scores.)
Answer:
Step-by-step explanation:
Since your population are the students in math class, you can use the z-score formula [tex]z=(x-\mu)/\sigma[/tex] in order to comparing the two math test scores. Where [tex]\mu [/tex] is the mean for the class, [tex]\sigma [/tex] is the standars deviation and x is Mary score.
For the first test [tex]\mu=.71 , \sigma=.18,x=.87[/tex] , so ,[tex]z_{1} = (.87-.71)/(.18)=.88[/tex].
For the second test [tex]\mu=.53 , \sigma=.14,x=.66[/tex] , so ,[tex]z_{1} = (.66-.53)/(.14)=.93[/tex]
Mary do better in the second test, relative to the rest of the class (because [tex].88 \leq .93[/tex], it means the second score is nearer to the mean score of the class than the first one )
Let P(x) denote the statement "2x+5 > 10." Which of the following is true?
P(0)
P(3)
P(2)
P(1)
Answer: P(3) is True
Step-by-step explanation:
The given statement is an inequality denoted as P(x). To find out which of the options is true you have to evaluate each given value of X in the inequality and perform the arithmetic operations, then you have to see if the expression makes sense.
For P(0): Replace X=0 in 2x+5>10
2(0)+5>10
0+5>10
5>10 is false because 5 is not greater than 10
For P(3): Replace X=3 in 2x+5>10
2(3)+5>10
6+5>10
11>10 is true because 11 is greater than 10
For P(2): Replace X=2 in 2x+5>10
2(2)+5>10
4+5>10
9>10 is false
For P(1): Replace X=1 in 2x+5>10
2(1)+5>10
2+5>10
7>10 is false
Jane is 20 lbs heavier than Alive. If Jane's weight is 120% that of Alice, find the weight of Alice
Answer:
Alice's weight is 100 lbs.
Step-by-step explanation:
Let's denote Jane's weight by J, and Alice's weight by A.
The exercise says that Jane is 20 lbs heavier than Alice. So that if you add 20 lbs to Alice's weight, you get Jane's weight. In equation form:
[tex]20+A=J[/tex]
It also mentions that Jane's weight is 120% that of Alice. So that if you multiply Alice's weight by 1.2, you get Jane's weight. In equation form:
[tex]1.2 \cdot A=J[/tex]
Plugging this second equation onto the first equation, we get:
[tex]20+A=1.2A[/tex]
And now solving for A:
[tex]20=1.2A-A[/tex]
[tex]20=0.2A[/tex]
[tex]\frac{20}{0.2}=A[/tex]
[tex]100=A[/tex]
Therefore Alice's weight is 100 lbs.
Convert 120 kW to W? MW?
Answer: 120000 W and 0.12 MW
Step-by-step explanation:
The expression 120 kW uses a metric prefix "k" (kilo) which is the same as multiply by 1000. So you can replace k by 1000 to convert the expression to the unit W.
120 kW= 120(1000) W= 120000 W.
To convert 120kW to MW, where the prefix M (mega) is equivalent to 1000000, you can use a conversion factor like (1 MW / 1000 kW) and multiply the expression by it.
Notice that (1 MW / 1000 kW) = 1, so the expression remains unaltered.
Then,
120 kW (1 MW / 1000 kW) = 0.12 MW
The data were gathered by the following procedure, reported in the study. "Between January and June 1998, parents of children aged 2-16 years [...] that were seen as outpatients in a university pediatric ophthalmology clinic completed a questionnaire on the child’s light exposure both at present and before the age of 2 years." Was this study observational, or was it a controlled experiment? Explain.
Answer:
This was an observational study.
Step-by-step explanation:
Given is that parents of children completed a questionnaire on the child’s light exposure both at present and before the age of 2 years.
This was an observational study since there is no treatment or control group.
We know that treatment, control groups or treatment groups are not specific to randomized control experiments.
The study where parents of pediatric ophthalmology patients completed questionnaires about light exposure is an observational study because data were collected without manipulating any variables.
Explanation:The study described in which parents filled out questionnaires about their children's light exposure is an example of an observational study, not a controlled experiment. In an observational study, researchers collect data without manipulating any variables. In this case, the researchers gathered information on light exposure by asking parents to complete a questionnaire, but they did not control or alter the children's light exposure themselves.
Unlike in an observational study, a controlled experiment involves actively manipulating one variable (the independent variable) to determine if it causes a change in another variable (the dependent variable), often comparing against a control group in a systematic way. An example of a controlled experiment includes the trial of Jonas Salk's polio vaccine, in which one group received the vaccine and another group received a placebo.
Part D 1 light-minute in miles Express your answer using three significant figures. ? mi 1 light minute Request Answer Submit
Answer:
1 light-minute ≈ 1.12×10⁷ miles, three significant figures.
Step-by-step explanation:
Light-second is equal to the distance traveled per second by light in space, which is equal to 299,792,458 metres. Other units used are light-day, light-hour and light-minute.
Significant figures are the figures or digits of a number that carry meaning and contribute to the precision of the given number.
1 light-minute = 1.118×10⁷ miles, has four significant figures.
To express this number in three significant figures, the given number is rounded.
1 light-minute = 1.118×10⁷ miles ≈ 1.12×10⁷ miles, has three significant figures, as the non-zero digits are significant.
A million years ago, an alien species built a vertical tower on a horizontal plane. When they returned they discovered that the ground had tilted so that measurements of 3 points on the ground gave coordinates of (0,0,0),(1,3,0)(0,0,0),(1,3,0), and (0,3,2)(0,3,2). By what angle does the tower now deviate from the vertical?
Answer:
The tower deviates [tex]64^\circ36'[/tex] from the vertical.
Step-by-step explanation:
Having 3 point of our new plane we can construct vectors on the plane by substacting 2 of them:
[tex]v_1=(0,3,2)-(0,0,0)\\v_2=(1,3,0)-(0,0,0)[/tex]
These vectors are on the plane, so a cross product between them will give us a vector perpendicular to the plane:
[tex](0,3,2)\times(1,3,0)=\left[\begin{array}{ccc}i&j&k\\0&3&2\\1&3&0\end{array}\right] =(-6,2,-3)[/tex]
Asuming that the aliens used our conventions, the original plane was perpendicular to the z axis, so that a perpendicular vector to that plane was
(0,0,1)
We know that a dot product between 2 vectors |V.W| = |V| |W| cos(α), where α is the angle between them. If we use the vector perpendicular to this plane, and the one perpendicular to the original plane, α will represent the deviation angle of our new plane.
[tex]\|(-6,2,-3)\|=7\\\|(0,0,1)\|=1[/tex]
[tex](-6,2,-3)\odot(0,0,1)= 7 cos(\alpha )\\\\ -3=7 cos(\alpha ) \\ \alpha=arc\ cos \frac{-3}{7} =115.37 ^\circ\\[/tex]
Since this angle is greater than 90 degrees it means that the vector we calculated as perpendicular to the plane points towards negative z (this can be seen by the -3 z component)
To fix this we can calculate a new perpendicular vector, or simply compare ir with the vector (0,0,-1). The latter is easier:
[tex](-6,2,-3)\odot(0,0,-1)= 7 cos(\alpha )\\\\ 3=7 cos(\alpha ) \\ \alpha=arc\ cos \frac{3}{7} =64.6^\circ =64^\circ36'[/tex]
In an arithmetic sequence, the nth term an is given by the formula an=a1+(n−1)d, where a1 is the first term and d is the common difference. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1, where r is the common ratio. Use these formulas to determine the indicated term in the given sequence.
The 10th term of 40,10, 5/2, 5/8, ....
Answer:
The 10th term of given sequence is [tex]\frac{5}{32768}[/tex].
Step-by-step explanation:
The given sequence is
[tex]40,10, \frac{5}{2}, \frac{5}{8}, ....[/tex]
The given sequence is a geometric sequence because it have common ratio.
[tex]r=\frac{10}{40}=\frac{\frac{5}{2}}{10}=\frac{\frac{5}{8}}{\frac{5}{2}}=\frac{1}{4}[/tex]
In the given sequence the first term of the sequence is 40.
[tex]a_1=40[/tex]
The nth term of a GP is
[tex]a_n=a_1r^{n-1}[/tex]
where, [tex]a_1[/tex] is first term and r is common ratio.
Substitute [tex]a_1=40[/tex] and [tex]r=\frac{1}{4}[/tex] in the above formula.
[tex]a_n=40(\frac{1}{4})^{n-1}[/tex]
Substitute n=10 , to find the 10th term.
[tex]a_{10}=40(\frac{1}{4})^{10-1}[/tex]
[tex]a_{10}=\frac{5}{32768}[/tex]
Therefore the 10th term of given sequence is [tex]\frac{5}{32768}[/tex].