Answer:
The initial population was approximatedly 3535 inhabitants.
Step-by-step explanation:
The population of the city can be given by the following differential equation.
[tex]\frac{dP}{dt} = Pr[/tex],
In which r is the rate of growth of the population.
We can solve this diffential equation by the variable separation method.
[tex]\frac{dP}{dt} = Pr[/tex]
[tex]\frac{dP}{P} = r dt[/tex]
Integrating both sides:
[tex]ln P = rt + c[/tex]
Since ln and the exponential are inverse operations, to write P in function of t, we apply ln to both sides.
[tex]e^{ln P} = e^{rt + C}[/tex]
[tex]P(t) = Ce^{rt}[/tex]
C is the initial population, so:
[tex]P(t) = P(0)e^{rt}[/tex]
Now, we apply the problem's statements to first find the growth rate and then the initial population.
The problem states that:
In two years the population has doubled:
[tex]P(2) = 2P(0)[/tex]
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]2P(0) = P(0)e^{2r}[/tex]
[tex]2 = e^{2r}[/tex]
To isolate r, we apply ln both sides
[tex]e^{2r} = 2[/tex]
[tex]ln e^{2r} = ln 2[/tex]
[tex]2r = 0.69[/tex]
[tex]r = \frac{0.69}{2}[/tex]
[tex]r = 0.3466[/tex]
So
[tex]P(t) = P(0)e^{0.3466t}[/tex]
In two years the population has doubled and a year later there were 10,000 inhabitants.
[tex]P(3) = 10,000[/tex]
[tex]P(t) = P(0)e^{0.3466t}[/tex]
[tex]10,000= P(0)e^{0.3466*3}[/tex]
[tex]P(0) = \frac{10,000}{e^{1.04}}[/tex]
[tex]P(0) = 3534.55[/tex]
The initial population was approximatedly 3535 inhabitants.
Give the equivalent measure of the ff. use the international metric system. show your solution a. 2.5km = __mm b. 0.05cm= ___mm c. 200.5 g = __kg d 0.03 t = __g e. 30412 sec = __hr
Answer and Explanation:
To find : Convert the given units ?
Solution :
a) 2.5 km to mm
[tex]1\ km = 1000000\ mm[/tex]
[tex]2.5\ km = 2.5\times 1000000\ mm[/tex]
[tex]2.5\ km = 2500000\ mm[/tex]
b) 0.05 cm to mm
[tex]1\ cm = 10\ mm[/tex]
[tex]0.05\ cm =0.05\times 10\ mm[/tex]
[tex]0.05\ cm =0.5\ mm[/tex]
c) 200.5 g to kg
[tex]1\ g = 0.001\ kg[/tex]
[tex]200.5\ g =200.5\times 0.001\ kg[/tex]
[tex]200.5\ g =0.2005\ kg[/tex]
d) 0.03 tone into g
[tex]1\ t =1000000\ g[/tex]
[tex]0.03\ t =0.03\times 1000000\ g[/tex]
[tex]0.03\ t =30000\ g[/tex]
e) 3.0412 sec into hour
[tex]1\ sec =\frac{1}{3600}\ hr[/tex]
[tex]3.0412\ sec =3.0412\times \frac{1}{3600}\ hr[/tex]
[tex]3.0412\ sec =0.000844\ hr[/tex]
Use mathematical induction to prove that for each integer n ≥ 4, 5^n ≥ 2 2^n+1 + 100
The given Statement which we have to prove using mathematical induction is
[tex]5^n\geq 2*2^{n+1}+100[/tex]
for , n≥4.
⇒For, n=4
LHS
[tex]=5^4\\\\5*5*5*5\\\\=625\\\\\text{RHS}=2.2^{4+1}+100\\\\=64+100\\\\=164[/tex]
LHS >RHS
Hence this statement is true for, n=4.
⇒Suppose this statement is true for, n=k.
[tex]5^k\geq 2*2^{k+1}+100[/tex]
-------------------------------------------(1)
Now, we will prove that , this statement is true for, n=k+1.
[tex]5^{k+1}\geq 2*2^{k+1+1}+100\\\\5^{k+1}\geq 2^{k+3}+100[/tex]
LHS
[tex]5^{k+1}=5^k*5\\\\5^k*5\geq 5 \times(2*2^{k+1}+100)----\text{Using 1}\\\\5^k*5\geq (3+2) \times(2*2^{k+1}+100)\\\\ 5^k*5\geq 3\times (2^{k+2}+100)+2 \times(2*2^{k+1}+100)\\\\5^k*5\geq 3\times(2^{k+2}+100)+(2^{k+3}+200)\\\\5^{k+1}\geq (2^{k+3}+100)+3\times2^{k+2}+400\\\\5^{k+1}\geq (2^{k+3}+100)+\text{Any number}\\\\5^{k+1}\geq (2^{k+3}+100)[/tex]
Hence this Statement is true for , n=k+1, whenever it is true for, n=k.
Hence Proved.
A chemist wants to mix a 22% acid solution with a 36% acid solution to get 28 L of a 26% acid solution. How many liters of the 22% solution and how many liters of the 36% solution should be mixed?
Answer:
20 L of 22% solution and 8 L of 36% solution
Step-by-step explanation:
Volume of 22% solution + volume of 36% solution = volume of 26% solution
x + y = 28
Acid in 22% solution + acid in 36% solution = acid in 26% solution
0.22x + 0.36y = 0.26(28)
0.22x + 0.36y = 7.28
Solve the system of equations using either elimination or substitution. I'll use substitution:
x = 28 − y
0.22(28 − y) + 0.36y = 7.28
6.16 − 0.22y + 0.36y = 7.28
0.14y = 1.12
y = 8
x = 28 − y
x = 20
The chemist should use 20 L of 22% solution and 8 L of 36% solution.
Answer:
There should be mixed 20 L of the 22% acid solution with 8L of the 36% acid solution
Step-by-step explanation:
We are mixing two acids.
x = liters of 22% acid solution
y = liters of 36% acid solution
x + y = 28 (total liters)
0.22x +0.36y = 0.26* 28
Since x+y=28 means y = 28-x
Now we will use substitution to find x
0.22x + 0.36(28-x) = 0.26 * 28
0.22x + 10.08 - 0.36x = 7.28
0.14x = 2.8
x = 20
y = 28 - 20 = 8
⇒ We use 20 liters of the 22% solution to be mixed with 8 liters of the 36% solution to form 28l of a 26% acid solution.
Estimate how many books can be shelved in a college library with 3500 m2 of floor space. Assume 8 shelves high, having books on both sides, with corridors 1.5 m wide. Assume books are approximately 25 cm deep and 5 cm wide, on average.
Approximately 347,200 books can be shelved in the college library with 3500 m² of floor space.
Given:
Floor space of the library = 3500 m²
Height of each shelf = 1.5 mWidth of each shelf = 0.5 m (assuming each book is 5 cm wide and there are two rows of books on each shelf)Depth of each shelf = 0.25 m (assuming each book is 25 cm deep)Width of each corridor = 1.5 mTo estimate how many books can be shelved in a college library with 3500 m² of floor space, we can use the following assumptions:
8 shelves high on both sidesCorridors 1.5 m wideBooks are approximately 25 cm deep and 5 cm wide, on average.To calculate the number of books that can be shelved, we need to find the volume of the shelving space and divide it by the volume of each book.
To find the volume of the shelving space, we need to subtract the volume of the corridors from the total volume of the library. The total volume of the library is:
[tex]V_{library} = (63\ yards) \times (32 \ yards) \times (6\ yards)[/tex]
Converting yards to meters, we get:
[tex]V_{library} = (63 \times 0.9144 \ meters) \times (32 \times 0.9144 \ meters) \times (6 \times 0.9144\ meters)[/tex]
Simplifying the equation, we get:
[tex]V_{library} \approx 1407\ m^3[/tex]
The volume of the corridors can be calculated as follows:
[tex]V_{corridors} = (8\ shelves) \times (0.5\ m + 1.5\ m) \times (1.5\ m) \times (63\ m + 32\ m)[/tex]
Simplifying the equation, we get:
[tex]V_{corridors} = 756\ m^3[/tex]
Therefore, the volume of the shelving space is:
[tex]V_{shelving} = V_{library} - V_{corridors} \\V_{shelving} \approx 651 \ m^3[/tex]
To find the volume of each book, we can multiply the depth, width, and height of each book:
[tex]V_{book} = (0.25\ m) \times (0.05\ m) \times (0.15\ m)[/tex]
Simplifying the equation, we get:
[tex]V_{book} = 0.001875 \ m^3[/tex]
Finally, we can divide the volume of the shelving space by the volume of each book to find the number of books that can be shelved:
[tex]Number\ of\ books = \frac{V_{shelving}} {V_{book}} \\Number\ of\ books \approx 347,200[/tex]
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The number of books can be shelved in a college library with [tex]3500 \ m^2[/tex] of floor space is 700,000 books.
Assume that the room is square-shaped.
The size of an average book is [tex]0.05\times0.25 \ m[/tex]. Thus, the thickness of the shelf is 0.5 m.
Area of the room, [tex]A=3500 \ m^2[/tex]
Width of the corridor space, [tex]W_c=1.5 \ m[/tex]
As the room is square-shaped, its width is as follows:
[tex]W_{room}=\sqrt{3500}[/tex]
[tex]= 59.16 m[/tex]
The area of a square with side a is [tex]a^2[/tex].
The total number of rows (r) is
[tex]r=\frac{W_{room}}{W_c+shelfsize}[/tex]
[tex]= \frac{59.16}{1.5+0.5}[/tex]
[tex]= 29.28 m[/tex]
The total number of shelves
[tex]= r\times 8(height of the shelves)\times2(facing both sides)[/tex]
[tex]= 473.28[/tex]
As the room is square-shaped, the length of each shelf is [tex]59.16 \ m[/tex].
The number of books on each shelf (n) is as follows:
[tex]n=\frac{Length \ of \ each \ row}{Thickness \ of \ each \ book}[/tex]
= [tex]\frac{59.16}{0.04}[/tex]
[tex]= 1479[/tex]
The total number of books is the sum of books on all shelves. Thus, the total number of books (N) is as follows:
[tex]N=n\times (Total \ number \ of \ shelves)[/tex]
[tex]= 1479\times473.28[/tex]
[tex]= 699981.12[/tex]
[tex]\approx 700000[/tex]
Thus, there are approximately 700,000 books.
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Let A 10,1,2,3,4,5,61, let B 0,1,2,3,4,5,6,7,81, and let R be the relation from A to B given by "the greatest common divisor of a and b is 2." [Note: "greatest common divisor" is sometimes called "highest common factor".] List the elements of R.
Answer:
R={(10,2),(10,4),(10,6),(2,2),(2,4),(2,6),(4,2),(4,6)}
Step-by-step explanation:
We are given that
A={10,1,2,3,4,5,61}
B={0,1,2,3,4,5,6,7,81}
We are given that R be the relation from A to B
R={gcd(a,b)=2,a[tex]\inA,b\inB[/tex]}
Gcd=Greatest common divisor of a and b.
We have to find the elements in R
(10,2)=2,(10,4)=2,(10,6)=2
(2,2)=2,(2,4)=2,(2,6)=2
(4,2)=2,(4,6)=2
Therefore, R={(10,2),(10,4),(10,6),(2,2),(2,4),(2,6),(4,2),(4,6)}
There are 360 people in my school. 15 take calculus, physics, and chemistry, and 15 don't take any of them. 180 take calculus. Twice as many students take chemistry as take physics. 75 take both calculus and chemistry, and 75 take both physics and chemistry. Only 30 take both physics and calculus. How many students take physics?
Answer:
150 students take physics.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the students that take calculus.
-The set B represents the students that take physics
-The set C represents the students that take chemistry.
-The set D represents the students that do not take any of them.
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of students that take only calculus, [tex]A \cap B[/tex] is the number of students that take both calculus and physics, [tex]A \cap C[/tex] is the number of students that take both calculus and chemistry and [tex]A \cap B \cap C[/tex] is the number of students that take calculus, physics and chemistry.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C), D[/tex]
There are 360 people in my school. This means that:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) + D = 360[/tex]
The problem states that:
15 take calculus, physics, and chemistry, so:
[tex]A \cap B \cap C = 15[/tex]
15 don't take any of them, so:
[tex]D = 15[/tex]
75 take both calculus and chemistry, so:
[tex]A \cap C = 75[/tex]
75 take both physics and chemistry, so:
[tex]B \cap C = 75[/tex]
30 take both physics and calculus, so:
[tex]A \cap B = 30[/tex]
Solution:
The problem states that 180 take calculus. So
[tex]a + (A \cap B) + (A \cap C) + (A \cap B \cap C) = 180[/tex]
[tex]a + 30 + 75 + 15 = 180[/tex]
[tex]a = 180 - 120[/tex]
[tex]a = 60[/tex]
Twice as many students take chemistry as take physics:
It means that: [tex]C = 2B[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 75 + 30 + 15[/tex]
[tex]B = b + 120[/tex]
-------------------------------
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]C = c + 75 + 75 + 15[/tex]
[tex]C = c + 165[/tex]
----------------------------------
Our interest is the number of student that take physics. We have to find B. For this we need to find b. We can write c as a function o b, and then replacing it in the equations that sums all the subsets.
[tex]C = 2B[/tex]
[tex]c + 165 = 2(b+120)[/tex]
[tex]c = 2b + 240 - 165[/tex]
[tex]c = 2b + 75[/tex]
The equation that sums all the subsets is:
[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) + D = 360[/tex]
[tex]60 + b + 2b + 75 + 30 + 75 + 15 + 15 = 360[/tex]
[tex]3b + 270 = 360[/tex]
[tex]3b = 90[/tex]
[tex]b = \frac{90}{3}[/tex]
[tex]b = 30[/tex]
30 students take only physics.
The number of student that take physics is:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]B = b + 75 + 30 + 15[/tex]
[tex]B = 30 + 120[/tex]
[tex]B = 150[/tex]
150 students take physics.
Final answer:
Using a Venn Diagram approach and the information given, we find that 45 students take physics at the school.
Explanation:
To find out how many students take physics at the school, we can use the Venn Diagram principle and the given data. We know that 15 students take calculus, physics, and chemistry together. Additionally, 180 students take calculus, and twice as many students take chemistry as take physics. With 75 students taking both calculus and chemistry, and another 75 taking both physics and chemistry, while only 30 take both physics and calculus, we can establish relationships and solve for the number of students taking each subject.
Let's denote the number of students taking physics as P. Then, the number of students taking chemistry would be 2P.
Total taking calculus and chemistry (C ∩ Ch) = 75Total taking calculus and physics (C ∩ P) = 30Total taking physics and chemistry (P ∩ Ch) = 75Total taking all three (C ∩ P ∩ Ch) = 15Students taking none = 15To avoid double counting, we must subtract those taking all three subjects once for each combination:
(C ∩ Ch) - (C ∩ P ∩ Ch) = 75 - 15 = 60 (only calculus and chemistry)(C ∩ P) - (C ∩ P ∩ Ch) = 30 - 15 = 15 (only calculus and physics)(P ∩ Ch) - (C ∩ P ∩ Ch) = 75 - 15 = 60 (only physics and chemistry)With twice as many students in chemistry as in physics, we can write the equation:
180 + 2P + P - (15 + 60 + 15 + 60) + 15 = 360
Solving for P:
P + 2P + 180 - 150 + 15 = 3603P = 360 - 45 - 1803P = 135P = 45Therefore, 45 students take physics at the school.
Hello, I asked help for this problem several times to no answer. I dont understand it and can use any help. Please, and thank you. please see the picture attached!
Check the picture below.
let's recall that a straight-line has 180°, and that sum of all interior angles in a triangle is also 180°.
Decide which of the following represent true statements about the nature of set. For any that are false, provide a specific example where the statement in question does not hold.
(a) If A1 ⊇ A2 ⊇ A3 ⊇ A4 ... are all sets containing an infinite number of elements, then the intersection n-1 An is infinite as well.
Answer:
If the intersection is finite the statement is true, but if the intersection is infinite the statement is false.
Step-by-step explanation:
From the statement of the problem I am not sure if the intersection is finite or infinite. Then, I will study both cases.
Let us consider first the finite case: [tex]A = \cap_{i=1}^{n}A_i[/tex]. Because the condition A1 ⊇ A2 ⊇ A3 ⊇ A4 ... we can deduce that the set [tex]A_n[/tex] is a subset of each set [tex]A_i[/tex] with [tex] i\leq n[/tex]. Thus,
[tex]\cap_{i=1}^{n}A_i = A_n[/tex].
Therefore, as [tex]A_n[/tex] is infinite, the intersection is infinite.
Now, if we consider the infinite intersection, i.e. [tex]A = \cap_{k=1}^{\infty}A_k[/tex] the reasoning is slightly different. Take the sets
[tex]A_k = (0,1/k)[/tex] (this is, the open interval between 0 and [tex]1/k[/tex].)
Notice that (0,1) ⊇ (0,1/2) ⊇ (0, 1/3) ⊇(0,1/4) ⊇...So, the hypothesis of the problem are fulfilled. But,
[tex]\cap_{k=1}^{\infty}(0,1/k) = \empyset[/tex]
In order to prove the above statement, choose a real number [tex]x[/tex] between 0 and 1. Notice that, no matter how small [tex]x[/tex] is, there is a natural number [tex]K[/tex] such that [tex]1/K<x[/tex]. Then, the number [tex]x[/tex] is not in any interval [tex](0,1/k)[/tex] with [tex]k>K[/tex]. Therefore, [tex]x[/tex] is not in the set [tex]\cap_{k=1}^{\infty}(0,1/k)[\tex].
The investors club invests 500 at 6% simple interest. How much
is in the account afte 90 days?
Answer: There would be $507.39 in the account after 90 days.
Step-by-step explanation:
Since we have given that
Principal = $500
Rate of interest = 6%
Number of days = 90 days
As we know that "Simple interest":
[tex]Interest=\dfrac{P\times R\times T}{100}\\\\Interest=\dfrac{500\times 6\times 90}{100\times 365}\\\\Interest = \$7.39[/tex]
So, Amount = Principal + Interest
Amount = $500 + $ 7.39
Amount = $507.39
Hence, There would be $507.39 in the account after 90 days.
Determine whether the data shows a linear relationship. If so, write an equation of a line of it.
Answer:
The data do not show a linear relationshipStep-by-step explanation:
A plot of the points connected by straight lines makes it pretty clear they do not all fall on the same line. There is no linear relationship here.
Draw a graph of order 4 and size equal to zero. What is this graph called?
Answer:
The draw in the file is a realization of a graph of order 4 and size zero.
In the book of Douglas West, Introduction to Graph Theory the name of this graph is 'Trivial graph'
Step-by-step explanation:
Remember that the order of a graph is the number of vertices and the size of the graph is the number of edges of the graph.
Solve the system of linear equations using the Gauss-Jordan elimination method. 5x + 3y = 16 −2x + y = −13 (x, y) =
Answer:
The solution for this system is [tex]x = 5, y = 3[/tex].
Step-by-step explanation:
The Gauss-Jordan elimination method is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.
We have the following system:
[tex]5x + 3y = 16[/tex]
[tex]-2x + y = -13[/tex]
This system has the following augmented matrix.
[tex]\left[\begin{array}{ccc}5&3&16\\-2&1&-13\end{array}\right][/tex]
The first step is dividing the first line by 5. So:
[tex]L_{1} = \frac{L_{1}}{5}[/tex]
We now have
[tex]\left[\begin{array}{ccc}1&\frac{3}{5}&\frac{16}{5}\\-2&1&-13\end{array}\right][/tex]
Now i want to reduce the first row, so I do:
[tex]L_{2} = L_{2} + 2L_{1}[/tex]
So we have
[tex]\left[\begin{array}{ccc}1&\frac{3}{5}&\frac{16}{5}\\0&\frac{11}{5}&-\frac{33}{5}\end{array}\right][\tex].
Now, the first step to reduce the second row is:
[tex]L_{2} = \frac{5L_{2}}{11}[/tex]
So we have:
[tex]\left[\begin{array}{ccc}1&\frac{3}{5}&\frac{16}{5}\\0&1&-3\end{array}\right][/tex].
Now, to reduce the second row, we do:
[tex]L_{1} = L_{1} - \frac{3L_{2}}{5}[/tex]
And the augmented matrix is:
[tex]\left[\begin{array}{ccc}1&0&5\\0&1&-3\end{array}\right][/tex]
The solution for this system is [tex]x = 5, y = 3[/tex].
Answer:
This is it:
Step-by-step explanation:
don't click any links
Let a, b, c and x elements in the group G. In each of the following solve for x in terms of a, b, and c.
Solve axb =c
Answer:
The answer is [tex]x=a^{-1}cb^{-1}[/tex].
Step-by-step explanation:
First, it is important to recall that the group law is not commutative in general, so we cannot assume it here. In order to solve the exercise we need to remember the axioms of group, specially the existence of the inverse element, i.e., for each element [tex]g\in G[/tex] there exist another element, denoted by [tex]g^{-1}[/tex] such that [tex]gg^{-1}=e[/tex], where [tex]e[/tex] stands for the identity element of G.
So, given the equality [tex] axb=c [/tex] we make a left multiplication by [tex]a^{-1}[/tex] and we obtain:
[tex]a^{-1}axb =a^{-1}c. [/tex]
But, [tex]a^{-1}axb = exb = xb[/tex]. Hence, [tex]xb = a^{-1}c[/tex].
Now, in the equality [tex]xb = a^{-1}c[/tex] we make a right multiplication by [tex]b^{-1}[/tex], and we obtain
[tex] xbb^{-1} = a^{-1}cb^{-1}[/tex].
Recall that [tex]bb^{-1}=e[/tex] and [tex]xe=x[/tex]. Therefore,
[tex]x=a^{-1}cb^{-1}[/tex].
add the numbers in the series 3+11+19+27+.....+395+403.
Answer:
Sum = 10,232
Step-by-step explanation:
The given sequence is Arithmetic Progression.
Arithmetic Progression is a sequence in which every two neighbor digits have equal distances.
For finding the sum of given series firstly we find the number of terms in given series.
For finding the nth term, we use formula
aₙ = a + (n - 1) d
where, aₙ = value of nth term
a = First term
n = number of term
d = difference
Now, In given sequence: 3+11+19+27+.....+395+403
a = 3, d = 8, aₙ = 403
∴ 403 = 3 + (n - 1) × 8
⇒ n = 51
Now, the sum of series is determined by formula,
Sₙ = n ÷ 2 [ a + l]
where l = last term
⇒ Sₙ = 51 ÷ 2 [ 3 + 403]
⇒ Sₙ = 51 × 203
⇒ Sₙ = 10,232
Final answer:
The series 3+11+19+27+.....+395+403 is an arithmetic sequence with a common difference of 8. It has 51 terms, and the sum can be found using the formula Sn = n(a1 + an) / 2, which gives us a sum of 10353.
Explanation:
To add the numbers in the series 3+11+19+27+.....+395+403, we need to recognize that this series is an arithmetic sequence, where each term increases by a common difference. In this case, the common difference is 8 (since 11 - 3 = 8, 19 - 11 = 8, and so on).
First, we need to find the number of terms in the series. We know that an arithmetic series can be expressed as ann = a1 + (n - 1)d, where an is the nth term, a1 is the first term, d is the common difference, and n is the number of terms. Plugging in the values we have:
403 = 3 + (n - 1)8
400 = (n - 1)8
n - 1 = 50
n = 51
So there are 51 terms in the series. The sum of an arithmetic series is given by Sn = n(a1 + an) / 2. Plugging in the values we found:
S51 = 51(3 + 403) / 2
S51 = 51(406) / 2
S51 = 10353
Therefore, the sum of the series is 10353.
What is the negation of the following statement: "I exercise and I feel tired."
A. I exercise and I feel tired.
B. I exercise and I feel envigorated.
C. I don't exercise or I feel tired.
D. I exercise or I feel tired.
E. I don't exercise and I feel tired.
F. I don't exercise or I feel envigorated.
G. I don't exercise and I feel envigorated.
H. I exercise or I feel envigorated.
Answer:
G. I don't exercise and I feel envigorated.
Step-by-step explanation:
In this sentence the I exercise and I feel tired you need to say in order tos ay the negation of this sentence would be:
I do not exercise nor feel tired, since there´s no option that says this, we can choose the one that says I don´t exercise, and envigorated is the opposite than tired, so the correct option would be I don´t exercise and I feel envigorated.
Final answer:
The correct negation of the statement "I exercise and I feel tired" is "I don't exercise or I feel envigorated," which corresponds to option F.
Explanation:
The negation of the compound statement "I exercise and I feel tired" involves negating both parts of the statement and changing the conjunction 'and' to 'or'. This is in line with De Morgan's laws which state that the negation of a conjunction is the disjunction of the negations. Therefore, the negated form of the statement would be "I don't exercise or I don't feel tired." From the given options, the one that best matches this structure is:
F. I don't exercise or I feel envigorated.
Envigorated is understood as the opposite of feeling tired in this context. So, statement F is the correct negation as it correctly captures the negation of both parts of the original statement.
In order to use a ladder safely, the angle that the ladder forms with the ground should not exceed 70 degree. If you have a ladder that is 16 ft. long, what is the maximum height on the wall of a building that the ladder will safely reach? Round your answer to the nearest thousandth.
Answer:
Maximum safe height can be reached by ladder = 15.03. ft
Step-by-step explanation:
Given,
Let's assume the maximum safe height of wall = h
angle formed between ladder and ground = 70°
length of ladder = 16 ft
From the given data, it can be seen that ladder will form a right angle triangle structure with the wall
So,from the concept of trigonometry,
[tex]Sin70^o\ =\ \dfrac{\textrm{maximum safe height of wall}}{\textrm{length of ladder}}[/tex]
[tex]=>Sin70^o\ =\ \dfrac{h}{16\ ft}[/tex]
[tex]=>\ h\ =\ 16\times Sin70^o[/tex]
=> h = 16 x 0.9396
=> h = 15.03 ft
So, the maximum safe height that can be reached by the ladder will be 15.03 ft.
In a certain region of the country it is known from
pastexperience that theprobability of selecting an adult over 40
yearsof age with cancer is 0.05. If the probability of a
doctorcorrectly diagnosing a person with cancer as having the
disease is0.78 and the probability of incorrectly diagnosing a
person withoutcancer as having the disease is .06, what is the
probability that aperson is diagnosed as having cancer?
Answer:
There is a 9.6% probability that a person is diagnosed as having cancer.
Step-by-step explanation:
In this problem, we have these following probabilities:
A 5% probability that an adult over 40 has cancer.
This also means that:
There is a 95% probability that an adult over 40 does not have cancer. (Since either the adult has cancer or does not have cancer, and the sum of the probabilities is 100%).
A 78% probability of a person that has cancer being diagnosed,
A 6% probability of a person that does not have cancer being diagnosed.
What is the probability that a person is diagnosed as having cancer?
[tex]P = P_{1} + P_{2}[/tex]
[tex]P_{1}[/tex] is the probability of those who have cancer being diagnosed. So it is 78% of 5%. So
[tex]P_{1} = 0.05*0.78 = 0.039[/tex]
[tex]P_{2}[/tex] is the probability of those who do not have cancer being diagnosed. So it is 6% of 95%. So
[tex]P_{1} = 0.06*0.95 = 0.057[/tex]
So
[tex]P = P_{1} + P_{2} = 0.039 + 0.057 = 0.096[/tex]
There is a 9.6% probability that a person is diagnosed as having cancer.
Let z≥2. What is the remainder of 10^z −1 divided by 4?
Answer:
The remainder is 3.
Step-by-step explanation:
We have to find out,
[tex]10^z-1(mod 4)=?\text{ where }z\geq 2[/tex]
If z = 2,
[tex]10^{2}-1=100-1=99[/tex]
∵ 99 ( mod 4 ) = 3,
Suppose,
[tex](10^{k}-1)(mod 4)=3\forall \text{ k is an integer greater than 2,}[/tex]
Now,
[tex](10^{k+1}-1) ( mod 4)[/tex]
[tex]= (10^k.10 - 10+9)(mod 4)[/tex]
[tex] = 10(mod 4)\times (10^k-1)(mod 4 ) + 9 ( mod 4)[/tex]
[tex]= (2\times 3)(mod 4) + 1[/tex]
[tex]=2+1[/tex]
[tex]=3[/tex]
Hence, our assumption is correct.
The remainder of [tex]10^z -1[/tex] divided by 4 is 3 where, z ≥ 2.
Sales of a certain MP3 players are approximated by the relationship S(x) = 4740x + 31,000(0 x 5) where S(x) denotes the number of MP3 players sold in year x (x = 0 corresponds to the year 2000). Find the number of MP3 players expected to be sold in 2002.
Answer:
40480 MP3 players expected to be sold in 2002.
Step-by-step explanation:
Sales of a certain MP3 players are approximated by the relationship;
[tex]S(x)=4740x+ 31000[/tex] (0≤x≤5)
(x = 0 corresponds to the year 2000)
That means 2002 corresponds to x = 2
Now substituting x = 2 in the expression.
[tex]S(x)=4740(2)+ 31000[/tex]
= [tex]9480+31000[/tex]
= 40480.
Hence, 40480 MP3 players expected to be sold in 2002.
Which image (A'B'C'D') of ABCD cannot be produced using only reflections? A. B. C. D.
Answer:
the answer is D
Step-by-step explanation:
Suppose that for a function f,f(2) is not defined. Also suppose that limx→2−f(x)=7 and limx→2+f(x)=7. Which, if any, of the following statements is false? a) limx→2f(x)=7 b) f has jump discontinuity at x = 2 c) If we re-define f so that f(2) = 7 then the new function will be continuous at x = 2 d) f has removable discontinuity at x = 2 e) All of the above statements are true.
All of the statements are true.
If the limit of a function f(x) at x = a is exist .
[tex]\lim_{x \to a+} f(x)= \lim_{x \to a-} f(x)=f(a)[/tex]
Given that,
[tex]\lim_{x \to 2-} f(x)= \lim_{x \to 2+} f(x)=7[/tex]
But f(2) is not defined.
It means that function f(x) has jump discontinuity at x = 2
A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph.
So that, Function f(x) has removable discontinuity at x = 2
Learn more:
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The false statement among the given options is that redefining f(2) to 7 will make the new function continuous at x = 2.
Explanation:The false statement among the options given is (c) If we re-define f so that f(2) = 7 then the new function will be continuous at x = 2.
The given information states that both the left and right limits as x approaches 2 are equal to 7, which suggests that the limit as x approaches 2 exists and is equal to 7. This means that option (a) lim x→2 f(x)=7 is true.
We know that f(2) is not defined in the original function, meaning there is a hole in the graph at x = 2. Therefore, option (d) f has a removable discontinuity at x = 2 is also true.
However, if we redefine f(2) = 7, the new function will still have a jump discontinuity at x = 2 since there will be a discontinuity between the values of f(2) before and after the redefinition. Therefore, option (c) is false.
So, the correct answer is (c) If we redefine f so that f(2) = 7 then the new function will be continuous at x = 2.
Learn more about Limits and Continuity here:https://brainly.com/question/30328478
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The number of bacteria in a flask grows according to the differential equation (dy)/(dt)= 0.06 y In this question, time is measured in hours and the number of bacteria, y, is measured in millions. The number of bacteria at time t = 0 is 4 million. Enter a formula for the number of bacteria at time t y = Click here to preview your answer. Incorrect: Your answer is incorrect. What is the value of the growth constant? Growth constant : per hour. How long does it take for the number of bacteria to double? (Enter your answer correct to two decimal places.) Doubling time : hours. How many million bacteria will be present after 9 hours have passed? (Enter your answer correct to one decimal place.) Number present after 9 hours : million.
Answer:
a) y = 4e^(0.06t)
b) 0.06
c) 11.55 hours
d) 6.9 million
Step-by-step explanation:
When the growth rate (millions per hour) is proportional to the number (millions), the relationship is exponential. The growth rate is the constant of proportionality.
a) Formula for y(t):
y = 4e^(0.06t)
__
b) The growth constant is 0.06, the multiplier of t in the exponential function. It is the constant of proportionality in the given differential equation:
y' = 0.06y.
__
c) The doubling time is found from ...
2 = e^(0.06t) . . . the multiplying factor is 2 to double the original number
ln(2) = 0.06t . . . . taking natural logs
ln(2)/0.06 = t ≈ 11.55 h . . . . doubling time
__
d) Put t=9 into the formula from part (a). After 9 hours, there will be ...
y(9) = 4e^(0.06·9) ≈ 6.9 . . . . million bacteria present
Answer:
y = 4e^(0.06t).
Step-by-step explanation:
dy/dt = 0.06y
Solving:
dy = 0.06y dt
dy/y = 0.06dt
Integrating both sides:
ln y = 0.06t + C
y = e^(0.06t + C)
y = Ae^(0.06t) where A is a constant.
At t = 0 , y = 4 million so
y = 4 = Ae^0 = A
So the formula is
y = 4e^(0.06t).
What is amount of interest I earned on $4,000 deposited in a savings account with 4% interest compounded annually after 4 years?
Give answer in US dollars and cents, rounded to the nearest cent. Do NOT enter "$" sign
Answer:
$ 679.43
Step-by-step explanation:
Since, the amount formula in compound interest is,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where,
P = initial amount,
r = annual rate of interest,
n = number of compounding periods in a year,
t = number of years,
Here, P = $ 4,000, r = 4% = 0.04, t = 4 years, n = 1,
So, the amount after 4 years would be,
[tex]=4000(1+0.04)^4\approx \$4679.43[/tex]
Hence, the amount of interest,
[tex]I=A-P=4679.43-4000 = \$ 679.43[/tex]
find the solution of cos(t) * f ` (t) = sin(t)
Answer:
The solution is [tex]f(t)=-\ln \left|\cos \left(t\right)\right|+C[/tex]
Step-by-step explanation:
We know that this ordinary differential equation (ODE) is separable if we can write F(x,y) = f(x)g(y) for some function f(x), g(x).
We can write this ODE in this way
[tex]cos(t) \cdot f'(t)=sin(t)\\f'(t)=\frac{sin(t)}{cos(t)}[/tex]
[tex]\mathrm{If\quad }f^{'} \left(x\right)=g\left(x\right)\mathrm{\quad then\quad }f\left(x\right)=\int g\left(x\right)dx[/tex]
[tex]f(t) =\int\limits{\frac{sin(t)}{cos(t)}} \, dt[/tex]
To solve this integral we need to follow this steps
[tex]\int \frac{\sin \left(t\right)}{\cos \left(t\right)}dt = \\\mathrm{Apply\:u-substitution:}\:u=\cos \left(t\right)\\\int \frac{\sin \left(t\right)}{u}dt \\\mathrm{And \:du=-sin(t)\cdot dt}\\\mathrm{so \>dt=\frac{du}{-sin(t)}}\\\int \frac{\sin \left(t\right)}{u}dt = -\int \frac{1}{u}du[/tex]
[tex]\mathrm{Use\:the\:common\:integral}:\quad \int \frac{1}{u}du=\ln \left(\left|u\right|\right)\\-ln|u|\\\mathrm{Substitute\:back}\:u=\cos \left(t\right)\\-\ln \left|\cos \left(t\right)\right|\\[/tex]
Add the constant of integration
[tex]f(t)=-\ln \left|\cos \left(t\right)\right|+C[/tex]
The equation for a parabola has the form y = ax^2 + bz + c, where a, b, and c are constants and aメ0. Find an equation for the parabola that passes through the points (-1,14), (2,-7), and (5, 8) Answer: y-
Answer:
a = 2, b = -9, c = 3
Step-by-step explanation:
Replacing x, y values of the points in the equation y = a*x^2 + b*x +c give the following:
(-1,14)
14 = a*(-1)^2 + b*(-1) + c
(2,-7)
-7 = a*2^2 + b*2 + c
(5, 8)
8 = a*5^2 + b*5 + c
Rearranging:
a - b + c = 14
4*a + 2*b + c = -7
25*a + 5*b + c = 8
This is a linear system of equations with 3 equations and 3 unknows. In matrix notation the system is A*x = b whith:
A =
1 -1 1
4 2 1
25 5 1
x =
a
b
c
b =
14
-7
8
Solving A*x = b gives x = Inv(A)*b, where Inv(A) is the inverse matrix of A. From calculation software (I used Excel) you get:
inv(A) =
0.055555556 -0.111111111 0.055555556
-0.388888889 0.444444444 -0.055555556
0.555555556 0.555555556 -0.111111111
inv(A)*b
2
-9
3
So, a = 2, b = -9, c = 3
A real estate agent has surveyed houses in several nearby zip codes in an attempt to put together a comparison for a new property that she would like to put on the market. The 583 houses she surveyed have a mean price of $176,678 with a standard deviation of $61,029. The mean house size is 1,676 square ft, with a standard deviation of 582 square ft. (Use 2 decimal places for the questions below.) Which is more unusual in this market: a house in that sells for $357,000 or a house with an area of 3,600 square ft?
Answer:
The house with an area 3,600 square feet is more unusual
Step-by-step explanation:
Given:
Number of houses surveyed = 583
Mean price = $176,678
Standard deviation = $61,029
Mean house size = 1,676 square ft
standard deviation = 582 square ft
Now,
the as z score = [tex]\frac{\textup{(X - mean )}}{\textup{standard deviation}}[/tex]
thus,
for selling value of $357,000
z score = [tex]\frac{\textup{(357,000 - 176,678 )}}{\textup{61,029}}[/tex]
or
z score = 2.95
and for house with an area 3,600 square feet
z score = [tex]\frac{\textup{(3600 - 1676)}}{\textup{582}}[/tex]
or
z score = 3.30
Hence, the house with an area 3,600 square feet is more unusual
Final answer:
To determine the more unusual house in the market, we calculate the z-scores. A house priced at $357,000 has a z-score of 2.95, while a house of 3,600 square feet has a z-score of 3.31. Therefore, the larger house size is more unusual.
Explanation:
To determine which house is more unusual in the given market, we need to calculate the number of standard deviations each value is from the mean, also known as the z-score. The z-score is calculated by taking the difference between the value and the mean, and then dividing by the standard deviation. For the price of the house, the z-score is calculated as follows:
Z = (Value - Mean) / Standard Deviation
For the $357,000 house price:
Z = ($357,000 - $176,678) / $61,029 = 2.95
For the 3,600 square ft house:
Z = (3,600 - 1,676) / 582 = 3.31
The house with an area of 3,600 square ft is 3.31 standard deviations away from the mean, whereas the $357,000 house price is 2.95 standard deviations away from the mean. Hence, the house with an area of 3,600 square ft is more unusual compared to the market's average.
what equation is graphed in this figure
Answer:
tatho you get the aswer
Step-by-step explanation:
all you have to do is add
Answer:
noice
Step-by-step explanation:
Write the negation of: If we lose electricity, then the data will be lost.
Write the negation of: If we lose electricity, then the data will be lost. Proposedsolution: PA- I have proposed PQ but I need a proper negation and proof tables.
Step-by-step explanation:
Consider the provided statement.
If we lose electricity, then the data will be lost.
We are need to write the negation of the above statement.
First divide the whole statement in two parts
Let us consider p = We lose electricity and q = The data will be lost.
The symbol use for negation is tilde [tex]\sim[/tex]
[tex]\sim(p\rightarrow q)[/tex]
[tex]p\wedge \sim q[/tex]
Represent T for true and F for False.
The required table is shown below:
p q [tex](p\wedge \sim q)[/tex]
F F F
F T F
T F T
T T F
Hence the required table is shown above.
A theater has 10 seats in the first row and 30 seats in the 6th row. How many seats are in the 11th row?
(It's ok if you give me the equation, that's all I need please and thank you)
Find the difference per row:
10 seats in the first row
30 seats in the sixth row:
30 -10 = 20 seats difference.
6-1 = 5 rows difference.
20 seats / 5 rows = 4 seats per row.
This means for every additional row, there are 4 more seats per row.
The equation would be:
Sn = S +(n-1)*d
Where d is the difference = 4
S = number of seats from starting row = 10
n = the number of rows wanted
S(11) = 10 + (11-1)*4
S(11) = 10 + 10*4
S(11) = 10 + 40
S(11) = 50
Check:
Row 6 = 30 seats
Row 7 = 30 + 4 = 34 seats
Row 8 = 34 + 4 = 38 seats
Row 9 = 38 + 4 = 42 seats
Row 10 = 42 + 4 = 46 seats
Row 11 = 46 + 4 = 50 seats.
What is the x-intercept of the linear equation y = 4x – 4
Answer: (1,0)
Step-by-step explanation: What is the x-intercept of the linear equation y = 4x – 4?
y = 4x - 4
x-intercept ⇒ y = 0
which means that we need to substitute the y by 0.
0 = 4x-4
4x = 4
x = 1
As it is a linear equation, 1st degree, there is only one point.
This way, the linear y = 4x - 4 intercept x on point (1,0)