The summand (R?) is missing, but we can always come up with another one.
Divide the interval [0, 1] into [tex]n[/tex] subintervals of equal length [tex]\dfrac{1-0}n=\dfrac1n[/tex]:
[tex][0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right][/tex]
Let's consider a left-endpoint sum, so that we take values of [tex]f(\ell_i)={\ell_i}^3[/tex] where [tex]\ell_i[/tex] is given by the sequence
[tex]\ell_i=\dfrac{i-1}n[/tex]
with [tex]1\le i\le n[/tex]. Then the definite integral is equal to the Riemann sum
[tex]\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3[/tex]
[tex]=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}[/tex]
The limit expression for the area under the curve y = x³ as n approaches infinity is; ¹/₄
What is the integral limit?The given definition is area A of the region S that lies under the graph of the continuous function which is the limit of the sum of the areas of approximating rectangles.
The expression for the area under the curve y = x³ from 0 to 1 as a limit is;
[tex]\lim_{n \to \infty} \Sigma^{n} _{i = 1} (\frac{i}{n})^{3} * \frac{1}{n} }[/tex]
From the expression above, when we factor out 1/n⁴, we will get;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \Sigma^{n} _{i = 1} \frac{i}{n} }[/tex]
This is further broken down to get;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } (\frac{n(n + 1)}{2} )^{2} } }[/tex]
This will be simplified to;
[tex]\lim_{n \to \infty} \frac{1}{n^{4} } \frac{(n^{4} + 2n^{3} + n^{2}) }{2}[/tex]
This would be simplified to;
[tex]\lim_{n \to \infty} \frac{1}{4} + \frac{1}{2n} + \frac{1}{4n^{2} }[/tex]
At limit of n approaches ∞, we have;
Limit = ¹/₄
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Ethan is playing in a soccer league that has 6 teams (including his team). Each team plays every other team twice during the regular season. The top two teams play in a final championship game after the regular season. In this league, how many soccer games will be played in all? 7.
Answer:
There are going to be 31 matches played in the soccer league.
Step-by-step explanation:
The soccer league has 6 teams, so if every team plays against the others twice, there are going to be played 30 matches:
-Team 1: v Team 2 (2), v Team 3 (2), v Team 4 (2), v Team 5 (2), v Team 6 (2)
-Team 2: v Team 3 (2), v Team 4 (2), v Team 5 (2), v Team 6 (2)
-Team 3: v Team 4 (2), v Team 5 (2), v Team 6 (2)
-Team 4: v Team 5 (2), v Team 6 (2)
-Team 5: v Team 6 (2)
-Team 6: -
If there is a final championship game after the 30 regular season matches, there are going to be 31 matches played in the league.
Suppose the lifetime of a computer memory chip may be modeled by a Gamma distribution. The average lifetime is 4 years and the variance is 16/3 years squared. What is the probability that such a chip will have a lifetime of less than 8 years?
Final answer:
To find the probability that a computer memory chip will have a lifetime of less than 8 years, we can use the properties of the Gamma distribution.
Explanation:
To find the probability that a computer memory chip will have a lifetime of less than 8 years, we can use the properties of the Gamma distribution. The average lifetime of the chip is given as 4 years, which corresponds to the mean. The variance is given as 16/3 years squared, which is equal to the mean squared.
Using these values, we can determine the shape and rate parameters of the Gamma distribution. The shape (α) is equal to the mean squared divided by the variance, which in this case is 16/3. The rate (β) is equal to the mean divided by the variance, which in this case is 4/(16/3).
To find the probability that the chip will have a lifetime of less than 8 years, we can calculate the cumulative distribution function (CDF) of the Gamma distribution with the shape and rate parameters we obtained.
Installment Buying TV Town sells a big screen smart HDTV for $600 down and monthly payments of $30 for the next 3 years. If the interest rate is 1.25% per month on the unpaid balance, find (a) the cost of the TV (b) the total amount of interest paid
Answer:
a) $ 1465.418
b) $ 214.582
Step-by-step explanation:
Since, the monthly payment formula of a loan is,
[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]
Where, PV is the principal amount of the loan,
r is the monthly rate,
n is the total number of months,
Here, P = $ 30, r = 1.25 % = 0.0125, n = 36 ( since, time is 3 years also 1 year = 12 months )
Substituting the values,
[tex]30=\frac{PV(0.0125)}{1-(1+0.0125)^{-36}}[/tex]
By the graphing calculator,
[tex]PV=865.418[/tex]
a) Thus, the cost of the TV = Down Payment + Principal value of the loan
= $ 600 + $ 865.418
= $ 1465.418
b) Now, the total payment = Monthly payment × total months
= 30 × 36
= $ 1080
Hence, the total amount of interest paid = total payment - principal value of the loan
= $ 1080 - 865.418
= $ 214.582.
Please help me with this
Answer:
Yes;Each side of triangle PQR is the same length as the corresponding side of triangle STU
Step-by-step explanation:
You can observe the sides of both triangles to see if this property holds
Lets check the length of AB
A(0,3) and B(0,-1)
[tex]AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\\\\\\\AB=\sqrt{(0-0)^2+(-1-3)^2} \\\\\\\\AB=\sqrt{0^2+-4^2} \\\\\\AB=\sqrt{16} =4units[/tex]
Now check the length of the corresponding side DE
D(1,2) and E(1,-2)
[tex]DE=\sqrt{(1-1)^2+(-2-2)^2} \\\\\\DE=\sqrt{0^2+-4^2} \\\\\\DE=\sqrt{16} =4units[/tex]
The side AB has the same length as side DE.This is also true for the remaining corresponding sides.
The principal at Apple Blossom High School decided to take her students on a field trip to the movie theater. A total of 250 people went on the trip. Adults paid $4.50 for a ticket and students paid $2.50 for a ticket. How many students and how many adults went to the movies if they paid a total of $805 at the movie theater?
Answer:
90 adults; 160 students
Step-by-step explanation:
Let "a" represent the number of adults who went. The number of students can be represented by (250-a). Then the total cost of tickets is ...
4.50a +2.50(250-a) = 805
2a + 625 = 805 . . . . . . simplify
2a = 180 . . . . . . . . . . . . subtract 625
a = 90 . . . . . . . . . . . . . . divide by 2
# of students = 250 -90 = 160
160 students and 90 adults went to the movies.
Answer:
the answer is 90 adults , and 160 students
Express the answers to the following calculations in scientific notation, using the correct number of significant figures. (a) 145.75 + (2.3 × 10−1) × 10 (b) 79,500 / (2.5 × 102) × 10 (c) (7.0 × 10−3) − (8.0 × 10−4) × 10 (d) (1.0 × 104) × (9.9 × 106) × 10
Each calculation has been evaluated, taken all significant figures into consideration, and results have been presented in scientific notation. Special attention was given to rules related to multiplying numbers in scientific notation.
Explanation:The given calculations require us to use scientific notation and proper treatment of significant figures. We are using the fundamentals of arithmetic with scientific notation, which is based on the rules of exponents. Each calculation is treated as follows:
(a) For the expression 145.75 + (2.3 × 10−1) × 10 = 147.05. This result has five significant figures, but to write numbers in scientific notation, we should round off to two significant figures as the lowest number of significant figures is 2 (in 2.3). Therefore, 147.05 becomes 1.47 × 10² in scientific notation. (b) For the expression 79,500 / (2.5 × 10²) × 10 = 3180. This is in turn expressed in scientific notation with three significant figures (since 2.5 has 3 significant figures) as 3.18 × 10³. (c) For the expression (7.0 × 10−3) − (8.0 × 10−4) × 10 = 6.2 × 10-3. Converting to scientific notation using two significant figures (based on original values), get 6.2 × 10⁻³. (d) For the expression (1.0 × 10⁴) × (9.9 × 10⁶) × 10 = 9.9 × 10¹¹ based on the rule of multiplying the numbers out front and adding up the exponents. Learn more about Scientific Notation and Significant Figures here:
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If the profit is $8000 and the profit % is 4%, what are net sales?
Answer:
8000/4*100 = $200'000
Step-by-step explanation:
6. Use the element method and proof by contradiction to prove that for any sets A,B and C, if B nCCA, then (C - A) n (B - A)=0.
Answer:
Suppose,
(C - A) ∩ (B - A) ≠ ∅
Let x is an element of (C - A) ∩ (B - A),
That is, x ∈ (C - A) ∩ (B - A),
⇒ x ∈ C - A and x ∈ B - A
⇒ x ∈ C, x ∉ A and x ∈ B, x ∉ A
⇒ x ∈ B ∩ C and x ∉ A
⇒ B ∩ C ⊄ A
But we have given,
B ∩ C ⊂ A
Therefore, our assumption is wrong,
And, there is no common elements in (C - A) and (B-A),
That is, (C - A) ∩ (B - A) = ∅
Hence proved...
Convert 141 to base seven.
Answer:
(141)₁₀→(241)₇
Step-by-step explanation:
(141)₁₀→(?)₇
for conversion of number from decimal to base 7 value we have to
factor 141 by 7
which is shown in the figure attached below.
from the attached figure we can clearly see that the colored digit will
give the conversion
we will write the digit from the bottom as shown in figure
(141)₁₀→(241)₇
I NEED THIS DONE IN AN EXCEL SPREADSHEET WITH SOLUTIONS
The following probabilities for grades in management science have been determined based on past records:
Grade Probability
A 0.1
B 0.2
C 0.4
D 0.2
F 0.10
The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on.
Determine the expected grade and variance for the course.
Answer:
Expected Grade=2 i.e., C
Variance=1.2
Step-by-step explanation:
[tex]Expected\ value=E\left [ x \right ]=\sum _{i=1}^{k} x_{i}p_{i}[/tex]
The x values are
A = 4
B = 3
C = 2
D = 1
F = 0
Probability of each of the events
P(4)=0.1
P(3)=0.2
P(2)=0.4
P(1)=0.2
P(0)=0.1
[tex]E\left [ x \right ]=4\times 0.1+3\times 0.2+2\times 0.4+1\times 0.2+0\times 0.1\\\therefore E\left [ x \right ]=2[/tex]
Variance
[tex]Var\left ( x\right)=E\left [ x^2 \right ]-E\left [ x \right ]^2[/tex]
[tex]E\left [ x^2 \right ]=4^2 \times 0.1+3^2 \times 0.2+2^2 \times 0.4+1^2 \times 0.2+0^2 \times 0.1\\\Rightarrow E\left [ x^2 \right ]=5.2\\E\left [ x \right ]^2=2^2=4\\\therefore Var\left ( x\right)=5.2-4=1.2\\[/tex]
Explain why a positive times a negative is a negative number.
Explanation:
This can be explained by thinking numbers on the number line as:
Lets take we have to multiply a positive number (say, 2) with a negative number say (-3)
2×(-3)
Suppose someone is standing at 0 on the number line and to go to cover -3 , the person moves 3 units in the left hand side. Since, we have to compute for 2×(-3), The person has to cover the same distance twice. At last, he will be standing at -6, which is a negative number.
A image is shown below to represent the same.
Thus, a positive times a negative is a negative number.
find the gcd and lcm of 20 and 56
By gcd, I think you mean gcf ( Greatest Common Factor).
To find the gcf find all the factors of each number:
Factors of 20: 1, 2, 4,5 ,10 , 20
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
The largest common factor is 4.
LCM = Least Common Multiple
This is the smallest number that both numbers divide into evenly
Find the prime factors of each number:
Prime factors of 20: 2 * 2 * 5
Prime factors of 56: 2 * 2 * 2 * 7
To find the LCM, multiply all prime factors the most number of times they occur.
In the prime factor of 56, 2 appears 3 times and 7 appears once.
In the prime factor of 20 5 appears once.
LCM = 2 * 2 *2 * 7 * 5 = 280
Solving Quadratic Equations by completing the square:
z^2 - 3z - 5 = 0
Answer:
[tex](z-\frac{3}{2} )^2-\frac{29}{4}[/tex]
Step-by-step explanation:
We are given the following quadratic equation by completing the square:
[tex]z^2 - 3z - 5 = 0[/tex]
Rewriting the equation in the form [tex]x^2+2ax+a^2[/tex] to get:
[tex]z^2 - 3z - 5+(-\frac{3}{2} )^2-(-\frac{3}{2} )^2[/tex]
[tex]z^2-3z+(-\frac{3}{2} )^2=(z-\frac{3}{2} )^2[/tex]
Completing the square to get:
[tex] ( z - \frac{ 3 } { 2 } )^ 2 - 5 - ( - \frac { 3 } { 2 } ) ^ 2[/tex]
[tex](z-\frac{3}{2} )^2-\frac{29}{4}[/tex]
Answer: [tex]z_1=4.19\\\\z_2=-1.19[/tex]
Step-by-step explanation:
Add 5 to both sides of the equation:
[tex]z^2 - 3z - 5 +5= 0+5\\\\z^2 - 3z = 5[/tex]
Divide the coefficient of [tex]z[/tex] by two and square it:
[tex](\frac{b}{2})^2= (\frac{3}{2})^2[/tex]
Add it to both sides of the equation:
[tex]z^{2} -3z+ (\frac{3}{2})^2=5+ (\frac{3}{2})^2[/tex]
Then, simplifying:
[tex](z- \frac{3}{2})^2=\frac{29}{4}[/tex]
Apply square root to both sides and solve for "z":
[tex]\sqrt{(z- \frac{3}{2})^2}=\±\sqrt{\frac{29}{4} }\\\\z=\±\sqrt{\frac{29}{4}}+ \frac{3}{2}\\\\z_1=4.19\\\\z_2=-1.19[/tex]
Problem 2 Consider three functions f, g, and h, whose domain and target are Z. Let fx)x2 g(x)=2x (a) Evaluate fo g(0) (b) Give a mathematical expression for f o g
Answer:
a) 0; b) 4[tex]x^{2}[/tex]
Step-by-step explanation:
a) To compute f o g (0), first evaluate g(x) for x=0 and then evaluate f for x=g(0).
[tex]f \circ g (0)=f(2 \cdot 0)=f(0)=0^2[/tex]
b) To compute a mathematical expression for f o g do the same but instead of 0 use x,
[tex]f \circ g (x) = f( 2 \cdot x)= (2 \cdot x )^2[/tex]
In the question, f(x) = x², g(x) = 2x. We need to determine the value of function f composed with function g at 0 (f o g(0)), and the general expression for f o g(x). f o g(0) = 0 and (f o g)(x) = 4x².
Explanation:To solve this problem, we first need to understand that 'f o g' denotes the composition of function f and function g, defined as (f o g)(x) = f(g(x)). In this case, function f(x) = x^2 and function g(x) = 2x.
(a) To evaluate f o g at 0, we substitute x = 0 into g(x), giving us g(0) = 2*0 = 0. Substituting g(0) into f(x), we get f(g(0)) = f(0) = 0. So, f o g(0) = 0.
(b) For a general form of f o g, we substitute g(x) = 2x into f(x), resulting in (f o g)(x) = f(2x) = (2x)^2 = 4x^2.
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What is another name for validity in qualitative research? a. objectivity b. bias c. trustworthiness d. reliability
Answer:
(d) reliability
Step-by-step explanation:
mostly we see that validity and reliability is the key aspects of all research they help in differentiation between good and bad research so both are very necessary aspects of any research so the another name for validity in quantity research is reliability.
so the reliability will be the correct answer
so option (d) will be correct option
A sample of 81 calculus students at a large college had a mean mathematics ACT score of 26 with a standard deviation of 6. Find a 95% confidence interval for the mean mathematics ACT score for all calculus students at this college.
Final answer:
The 95% confidence interval for the mean mathematics ACT score for all calculus students at this college is calculated to be between 24.693 and 27.307.
Explanation:
To calculate a 95% confidence interval for the mean mathematics ACT score for all calculus students at the college, we use the sample mean and standard deviation along with the z-score for a 95% confidence level. Since the sample size is large (n=81), we can use the z-distribution as an approximation for the t-distribution.
The formula for a confidence interval is:
Mean ± (z-score * (Standard Deviation / √n))
Here, the sample mean (μ) is 26, the standard deviation (s) is 6, and the sample size (n) is 81. For a 95% confidence level, the z-score is approximately 1.96.
Now we calculate the margin of error (ME):
ME = 1.96 * (6 / √81) = 1.96 * (6 / 9) = 1.96 * 0.6667 = 1.307
Therefore, the 95% confidence interval is
26 ± 1.307
Lower Limit = 26 - 1.307 = 24.693
Upper Limit = 26 + 1.307 = 27.307
The confidence interval is (24.693, 27.307).
We estimate with 95 percent confidence that the true population mean for the mathematics ACT score for all calculus students at the college is between 24.693 and 27.307.
help please? even if someone gave me the steps to figure the answer myself, that'd be great
Answer:
The height is 28.57 cm.
The surface area is 9,628 cm^2.
Step-by-step explanation:
I assume the cooler is shaped like a rectangular prism with length and width of the base given, and with an unknown height.
volume = length * width * height
First, we convert the volume from liters to cubic centimeters.
60 liters * 1000 mL/L * 1 cm^3/mL = 60,000 cm^3
Now we substitute every dimension we have in the formula and solve for height, h.
60,000 cm^3 = 60 cm * 35 cm * h
60,000 cm^3 = 2,100 cm^2 * h
h = (60,000 cm^3)/(2,100 cm^2)
h = 28.57 cm
The height is 28.57 cm.
Now we calculate the internal surface area.
total surface area = area of the bases + area of the 4 sides
SA = 2 * 60 cm * 35 cm + (60 cm + 35 cm + 60 cm + 35 cm) * 28.57 cm
SA = 9,628 cm^2
The surface area is 9,628 cm^2.
2)Whitney is shopping for party supplies. She finds a package of 10 plates, 16 napkins, and a package of 8 cups. What is the least number of packages of plates, napkins, and cups so that she has the same number of each? Write answer in sentence form.
Final answer:
To have the same number of plates, napkins, and cups, Whitney will need at least 80 of each.
Explanation:
To find the least number of packages of plates, napkins, and cups so that Whitney has the same number of each, we need to find the least common multiple (LCM) of 10, 16, and 8. The LCM is the smallest multiple that all three numbers have in common.
10 = 2 x 5, 16 = 2 x 2 x 2 x 2, and 8 = 2 x 2 x 2
We can identify the prime factors of each number and then multiply the highest factor from each number. In this case, the LCM is 2 x 2 x 2 x 2 x 5 = 80.
Therefore, Whitney will need at least 80 plates, 80 napkins, and 80 cups to have the same number of each.
An urn contains 11 numbered balls, of which 6 are red and 5 are white. A sample of 4 balls is to be selected. How many samples contain at least 3 red balls?
Answer:
The total number of samples that contain at least 3 red balls is 115.
Step-by-step explanation:
Total number of balls = 11
Total number of red balls = 6
Total number of white balls = 5
A sample of 4 balls is to be selected that contain at least 3 red. It means either 3 out of 4 balls are red or 4 out of 4 ball are red.
[tex]\text{Total ways}=\text{Three balls are red}+\text{Four balls are red}[/tex]
[tex]\text{Total ways}=^6C_3\times ^5C_1+^6C_4\times ^5C_0[/tex]
Combination formula:
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Using this formula we get
[tex]\text{Total ways}=\frac{6!}{3!(6-3)!}\times \frac{5!}{1!(5-1)!}+\frac{6!}{4!(6-4)!}\times \frac{5!}{0!(5-0)!}[/tex]
[tex]\text{Total ways}=20\times 5+15\times 1[/tex]
[tex]\text{Total ways}=115[/tex]
Therefore the total number of samples that contain at least 3 red balls is 115.
Using the combination formula, it is found that 115 samples contain at least 3 red balls.
The balls are chosen without replacement, which is why the combination formula is used.
Combination formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by:
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this problem, the outcomes with at least 3 red balls are:
3 red from a set of 6 and 1 white from a set of 5.4 red from a set of 6.Hence:
[tex]T = C_{6,3}C_{5,1} + C_{6,4} = \frac{6!}{3!3!}\frac{5!}{1!4!} + \frac{6!}{4!2!} = 20(5) + 15 = 100 + 15 = 115[/tex]
115 samples contain at least 3 red balls.
A similar problem is given at https://brainly.com/question/24437717
Rewrite the system of linear equations as a matrix equation AX = B.
leftbrace2.gif
2x1 + 5x2 − 2x3 + x4 + 2x5 = 1
x1 + x2 − 2x3 + x4 + 4x5 = 5
Answer:
Given: [tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]
[tex]2x_{1}+5 x_{2} - 2_{3} + x_{4} + 2x_{5}\\[tex]
[tex]x_{1}+ x_{2} - 2_{3} + x_{4} + 4x_{5}\\[/tex]
The system of linear equations in matrix form may be written as:
AX=B,
where,
A is coefficient matrix of order [tex]2\times 4[/tex] and is given by:
A = [tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]
X is variable matrix of order [tex]5\times 1[/tex] and is given by:
X= [tex]\begin{bmatrix}x_{_{1}}\\x_{_{2}}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}[/tex]
and B is the contant matrix of order [tex]2\times 1[/tex] and is given by:
B = [tex]\begin{bmatrix}1\\5\end{bmatrix}[/tex]
Now, AX=B
[tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]. [tex]\begin{bmatrix}x_{_{1}}\\x_{_{2}}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}[/tex] = [tex]\begin{bmatrix}1\\5\end{bmatrix}[/tex]
15. The formula for the surface area of a rectangular solid is S 2HW + 2LW + 2LH, where S, H, W, and L represent surface area, height, width, and length, respectively. Solve this formula for W.
Answer:
The answer is
[tex]W=\frac{S-2LH}{2H+2L}[/tex]
Step-by-step explanation:
The formula for the area of a solid rectangle is
[tex]S = 2HW+2LW+2LH[/tex]
Solve it for W
[tex]2HW+2LW=S-2LH\\\\W(2H+2L)=S-2LH\\\\W=\frac{S-2LH}{2H+2L}[/tex]
Mike deposited $850 into the bank in July. From July to December, the amount of money he deposited into the bank increased by 15% per month. What's the total amount of money in his account after December? Round your answer to the nearest dollar. Show your work. 4.
Answer:
$1.710
Step-by-step explanation:
Mike deposited $850 into the bank in July.
In August his balance will be: $850×1.15 = $977.5
In September his balance will be: $977.5×1.15 = $1124.125
In October his balance will be: $1124.125×1.15 = $1292.74375
In November his balance will be: $1292.74375×1.15 = $1.486,6553125
In December his balance will be: $1.486,6553125×1.15 = $1.709,653609375
Therefore, the amount of money he will have after december will be $1.710
Professor N. Timmy Date has 31 students in his Calculus class and 17 students in his Discrete Mathematics class.
(a) Assuming that there are no students who take both classes, how many students does Professor Date have?
(b) Assuming that there are five students who take both classes, how many students does Professor Date have?
Answer: a) 48
b) 43
Step-by-step explanation:
Given : The number of students Professor Date has in his Calculus class = 31
The number of students Professor Date has in his Discrete Mathematics class = 17
(a) If we assume that there are no students who take both classes, then the total number of students Professor Date Has = 31+17=48
(b) If we assume that there are five students who take both classes, then the total number of students Professor Date Has = 31+17-5=43
For each of the squences below, find a formula that generates the sequence.
(a) 10,20,10,20,10,20,10...
Answer:
[tex]a_{n}=15 + (-1)^n * 5[/tex]
Step-by-step explanation:
First, we notice that the when n is odd, [tex]a_{n}[/tex] = 10. And when n is even, [tex]a_{n}[/tex] = 20.
The average of 10 and 20 is [tex](10+20)/2 = 15[/tex]. So, the distance between 15 and 10 is the same that between 15 and 20.
That distance is 5.
From 15, we need to subtract 5 to get 10 when n is odd and we need to add 5 to get 20 when n is even.
The easiest way to express that oscilation is using [tex](-1)^n[/tex], because it is (-1) when n is odd and 1 when is even. And when multiplied by 5, it will add or subtract 5 as we wanted.
A sample of 100 wood and 100 graphite tennis rackets are taken from the warehouse. If 15 wood and 14 graphite are defective and one racket is randomly selected from the sample, find the probability that the racket is wood or defective.
Answer:
The probability that the racket is wood or defective is 0.57.
Step-by-step explanation:
Let W represents wood racket, G represents the graphite racket and D represents the defective racket,
Given,
n(W) = 100,
n(G) = 100,
⇒ Total rackets = 100 + 100 = 200
n(W∩D) = 15,
n(G∩D) = 14,
⇒ n(D) = n(W∩D) + n(G∩D) = 15 + 14 = 29,
We know that,
n(W∪D) = n(W) + n(D) - n(W∩D)
= 100 + 29 - 15
= 100 + 14
= 114,
Hence, the probability that the racket is wood or defective,
[tex]P(W\cup D) = \frac{114}{200}[/tex]
[tex]=0.57[/tex]
For the mathematics part of the SAT the mean is 514 with a standard deviation of 113, and for the mathematics part of the ACT the mean is 20.6 with a standard deviation of 5.1. Bob scores a 660 on the SAT and a 27 on the ACT. Use z-scores to determine on which test he performed better.
Answer:
Bob performed better in mathematics part of the SAT than the ACT
Step-by-step explanation:
We need to calculate the z-scores for both parts and compare them.
Z-score for the SAT is calculated using the formula:
[tex]Z=\frac{X-\mu}{\sigma}[/tex]
where [tex]\mu=514[/tex] is the mean and [tex]\sigma=113[/tex] is the standard deviation, and [tex]X=660[/tex] is the SAT test score.
We plug in these values to obtain:
[tex]Z=\frac{660-514}{113}[/tex]
[tex]Z=\frac{146}{113}=1.29[/tex] to the nearest hundredth.
We use the same formula to calculate the z-score for the ACT too.
Where [tex]\mu=20.6[/tex] is the mean and [tex]\sigma=5.1[/tex] is the standard deviation, and [tex]X=27[/tex] is the ACT test score.
We substitute the values to get:
[tex]Z=\frac{27-20.6}{5.1}=1.25[/tex] to the nearest hundredth.
Since 1.29 > 1.25, Bob performed better in mathematics part of the SAT
Using z-scores to determine where Bob's performance stands compared to others, we find that he performed slightly better on the SAT with a z-score of 1.29, than on the ACT with a z-score of 1.25.
Explanation:To determine on which test Bob performed better, we have to calculate his z-scores on both the SAT and ACT. Z-score is a statistical measurement that describes a score's relationship to the mean of a group of scores. It indicates how many standard deviations an element is from the mean.
Here is how you calculate the z-score: z = (X - μ) / σ, where X is the individual score, μ is the mean, and σ is the standard deviation.
We can apply this formula to both of Bob's scores. For the SAT: z = (660 - 514) / 113 ≈ 1.29 For the ACT: z = (27 - 20.6) / 5.1 ≈ 1.25
Comparing the z-scores, Bob's score is above the mean by 1.29 standard deviations on the SAT and by 1.25 standard deviations on the ACT. Therefore, Bob performed slightly better on the SAT than on the ACT when comparing his scores to other test takers.
Learn more about Z-Scores here:https://brainly.com/question/15016913
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For a lottery game, a player must match a sequence of three repeatable numbers, ranging from 0 to 9, in exact order. With a single ticket, what is the probability of matching the three winning numbers?
Answer:[tex]\frac{1}{1000}[/tex]
Step-by-step explanation:
For the lottery game three numbers must match in exact order
From 0 to 9 total 10 numbers are there
Therefore selecting exactly same numbers as of winner is
=[tex]^{10}C_1\times ^{10}C_1\times ^{10}C_1 [/tex]
Since numbers are repeatable therefore each time we have a choice of choosing 1 number out o 10
=[tex]10\times 10\times 10[/tex]
Probability of winning=[tex]\frac{1}{1000}[/tex]
In a lottery game where the player must match a repeatable sequence of three numbers ranging from 0-9, the probability of a single ticket having the winning sequence is 1/1000, or 0.1%.
Explanation:The subject of this problem is probability in Mathematics, specifically in a lottery context. Each digit in the sequence can be any number from 0 to 9. Because these numbers are repeatable, this means there are 10 possible numbers for each of the three digits in the sequence. Therefore, to find the total number of possible sequences, you multiply the ten options for the first digit by the ten for the second and the ten for the third, which comes to 10 * 10 * 10 = 1000 total possible sequences.
Since you are only looking for one specific sequence being the winning number, that means there’s only 1 favorable outcome out of 1000. Therefore, the probability of your ticket having the winning number sequence is 1/1000, or 0.001 in decimal form or 0.1% in percentage form.
Learn more about Probability here:https://brainly.com/question/22962752
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PLEASE HELP TRIG SEE ATTACHMENT WILL MARK BRAINLIEST
Answer:
sin Ф=3/√13
Cos Ф=2/√13
Tan Ф=3/2
Step-by-step explanation:
Let x=2
Let y=3
Let r be the length of line segment drawn from origin to the point
[tex]r=\sqrt{x^2+y^2}[/tex]
Find r
[tex]r=\sqrt{2^2+3^2} =\sqrt{4+9} =\sqrt{13}[/tex]
Apply the relationship for sine, cosine and tan of Ф where
r=hypotenuse
Sine Ф=length of opposite side÷hypotenuse
Sin Ф=O/H where o=3, hypotenuse =√13
sin Ф=3/√13
CosineФ=length of adjacent side÷hypotenuse
Cos Ф=A/H
Cos Ф=2/√13
Tan Ф=opposite length÷adjacent length
TanФ=O/A
Tan Ф=3/2
At the beginning of 1990, 21.7 million people lived in the metropolitan area of a particular city, and the population was growing exponentially. The 1996 population was 25 million. If this trend continues, how large will the population be in the year 2010
Final answer:
To calculate the population in the year 2010 based on exponential growth from 1990, use a growth rate factor and the known population figures from the given years.
Explanation:
Population Growth Calculation:
Determine the growth rate factor from 1990 to 1996: 25 million / 21.7 million = 1.152
Apply the growth rate to find the population in 2010: 21.7 million * (1.152)¹⁴ (14 years from 1996 to 2010) = 48.9 million
From a standard 52-card deck, how many eight-card hands consists of three cards of one denomination, three cards of another denomination, and two cards of a third denomination?
Answer:82,368
Step-by-step explanation:
Final answer:
The total number of eight-card hands formed from a standard 52-card deck where the hand consists of three cards of one denomination, three cards of another, and two cards of a third is 27,456. This is calculated by finding combinations of denominations and the specific cards within those denominations.
Explanation:
To answer the question of how many eight-card hands can be formed from a standard 52-card deck, where the hand consists of three cards of one denomination, three cards of another denomination, and two cards of a third denomination, we have to use combinations. This is a problem of combinatorics in which we are finding the number of ways to select items from a group without regard to order.
First, we choose the denominations. There are 13 denominations and we want to choose 3 of them for our hand. Using the combination formula, this can be done in C(13, 3) ways:
C(13, 3) = 13! / (3! × (13-3)!) = 286
After choosing the denominations, we need to select the specific cards. For each of the first two denominations, we select 3 out of the 4 available suits, and for the third denomination, we select 2 out of the 4 suits. This can be done as follows:
C(4, 3) for the first denomination: C(4, 3) = 4 ways
C(4, 3) for the second denomination: C(4, 3) = 4 ways
C(4, 2) for the third denomination: C(4, 2) = 6 ways
Multiplying the ways to choose the denominations by the ways to choose the cards for each denomination gives us the total number of distinct hands.
Total hands = C(13, 3) × C(4, 3) × C(4, 3) × C(4, 2) = 286 × 4 × 4 × 6 = 27,456
Therefore, there are 27,456 different eight-card hands that meet the given criteria.