Answer:
[tex]x\ =\ \dfrac{209}{30}[/tex]
[tex]y\ =\ \dfrac{29}{18}[/tex]
[tex]z\ =\ \dfrac{64}{15}[/tex]
Step-by-step explanation:
Given equations are
15x + 15y + 10z = 106
5x + 15y + 25z = 135
15x + 10y - 5z = 42
The augmented matrix by using above equations can be written as
[tex]\left[\begin{array}{ccc}15&15&10\ \ |106\\5&15&25\ \ |135\\15&10&-5|42\end{array}\right][/tex]
[tex]R_1\ \rightarrow\ \dfrac{R_1}{15}[/tex]
[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\5&15&25|135\\15&15&-5|42\end{array}\right][/tex]
[tex]R_1\rightarrowR_2-5R1\ and\ R_3\rightarrow\ R_3-15R_1[/tex]
[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&10&\dfrac{65}{3}|\dfrac{299}{3}\\\\0&0&-15|-64\end{array}\right][/tex]
[tex]R_2\rightarrow\ \dfrac{R_2}{10}[/tex]
[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&-15|-64\end{array}\right][/tex]
[tex]R_3\rightarrow\ \dfrac{R_3}{-15}[/tex]
[tex]=\ \left[\begin{array}{ccc}1&1&\dfrac{10}{15}|\dfrac{106}{15}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]
[tex]R_1\rightarrow\ R_1-R_2[/tex]
[tex]=\ \left[\begin{array}{ccc}1&0&\dfrac{-3}{2}|\dfrac{17}{30}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]
[tex]R_1\rightarrow\ R_1+\dfrac{3}{2}R_3[/tex]
[tex]=\ \left[\begin{array}{ccc}1&0&0|\dfrac{209}{30}\\\\0&1&\dfrac{65}{30}|\dfrac{299}{30}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]
[tex]R_2\rightarrow\ R_2-\dfrac{65}{30}R_3[/tex]
[tex]=\ \left[\begin{array}{ccc}1&0&\0|\dfrac{209}{30}\\\\0&1&0|\dfrac{29}{18}\\\\0&0&1|\dfrac{64}{15}\end{array}\right][/tex]
Hence, we can write from augmented matrix,
[tex]x\ =\ \dfrac{209}{30}[/tex]
[tex]y\ =\ \dfrac{29}{18}[/tex]
[tex]z\ =\ \dfrac{64}{15}[/tex]
A hemispherical bowl of radius 12 inches is filled to a depth of h inches, where 0less than or equalshless than or equals12. Find the volume of water in the bowl as a function of h. (Check the special cases hequals0 and hequals12.)
Answer:
Suppose the bowl is situated such that the rim of the bowl touches the x axis, and the semicircular cross section of the bowl lies below the x-axis (in (iii) and (iv) quadrant ). Then the equation of the cross section of the bowl would be [tex]x^2+y^2=144[/tex], where y≤ 0,
⇒ [tex]y=-\sqrt{144-x^2}[/tex]
Here, h represents the depth of water,
Thus, by using shell method,
The volume of the disk would be,
[tex]V(h) = \pi \int_{-12}^{-12+h} x^2 dx[/tex]
[tex]= \pi \int_{-12}^{-12+h} (144-y^2) dy[/tex]
[tex]= \pi |144y-\frac{y^3}{3}|_{-12}^{-12+h}[/tex]
[tex]=\pi [ (144(-12+h)-\frac{(-12+h)^3}{3}-144(-12)+\frac{(-12)^3}{3}}][/tex]
[tex]=\pi [ -1728 + 144h - \frac{1}{3}(-1728+h^3+432h-36h^2)+1728-\frac{1728}{3}][/tex]
[tex]=\pi [ 144h - \frac{1}{3}(h^3+432h-36h^2}{3}][/tex]
[tex]=\pi [ 144h - \frac{h^3}{3} - 144h + 12h^2][/tex]
[tex]=\pi ( 12h^2 - \frac{h^3}{3})[/tex]
Special cases :
If h = 0,
[tex]V(0) = 0[/tex]
If h = 12,
[tex]V(12) = \pi ( 1728 - 576) = 1152\pi [/tex]
How many ways can you make change for a $50 bill using $5, $10, and $20 bills?
Answer:
There are 12 ways to make change for a $50 bill using $5, $10 and $20 bills
Step-by-step explanation:
Let's write down every possibility starting by using the largest quantity of $20 bills and we'll go from there, everytime that we get a $10 bill we will split it in the next option into 2 $5 bills.
(20)(20)(10)
(20)(20)(5)(5)
(20) (10)(10)(10)
(20)(10)(10)(5)(5)
(20)(10)(5)(5)(5)(5)
(20)(5)(5)(5)(5)(5)(5)
Now we start with the largest quantity of $10 bills (5) and go from there, splitting them into two 5 dollar bills in the next option.
(10)(10)(10)(10)(10)
(10)(10)(10)(10)(5)(5)
(10)(10)(10)(5)(5)(5)(5)
(10)(10)(5)(5)(5)(5)(5)(5)
(10)(5)(5)(5)(5)(5)(5)(5)(5)
(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)
Answer:
12 ways
Step-by-step explanation:
20 x 20 x 10 x 20 x 20 x 5 x 5
Prove that x-1 is a factor of x^n-1 for any positive integer n.
Answer:
[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]
Step-by-step explanation:
[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]
We will prove this with the help of principal of mathematical induction.
For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.
Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].
Thus, [tex]x^k - 1[/tex] can be written equal to [tex]y(x-1)[/tex], where y is an integer.
Now, we will prove that the given statement is true for n = k+1
[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]
Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].
Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.
If a is an integer, prove that (14a +3,21a + 4) 1
Answer:
See explanation below.
Step-by-step explanation:
To prove that the greatest common divisor of two numbers is 1, we use the Euclidean algorithm.
1. In this case, and applying the algorithm we would have:
(14a + 3, 21a + 4) = (14a + 3, 7a + 1) = (1, 7a + 1) = 1
2. Other way of proving this statement would be that we will need to find two integers x and y such that 1 = (14a + 3) x + (21a + 4) y
Let's make x = 3 and y = -2
Then we would have:
[tex](14a+3)(3) + (21a+4)(-2)\\=42a+9-42a-8\\=1[/tex]
Therefore, (14a + 3, 21a + 4) = 1
Use a traditional clock face to determine the next three terms in the following sequence 2.9.4. 11.6.... The next three terms, in order, are and melhores de
Answer:
The next three terms are 1, 8 and 3.
Step-by-step explanation:
Consider the provided sequence,
2, 9, 4, 11, 6....
We need to find the next three terms.
It is given that we need to use a traditional clock face to determine the next three terms in the following sequence.
In the above sequence we are asked to add 7 hours to each time on the traditional clock face.
2 + 7 = 9
9 + 7 = 16 In traditional clock 16 is 4 O'clock
4 + 7 = 11
11 + 7 = 18 In traditional clock 18 is 6 O'clock
6 + 7 = 13 In traditional clock 13 is 1 O'clock
1 + 7 = 8
8 + 7 = 15 In traditional clock 15 is 3 O'clock
Hence, the next three terms are 1, 8 and 3.
the forecast calls for a 30% chance of snow today and a 40% chance of snow tomorrow. what are the chances it will rain two days in a row?
Answer:
The chances it will rain two days in a row is 12%
Step-by-step explanation:
The forecast calls for a 30% chance of snow today
So, chance of snowfall today = 30% = 0.3
A 40% chance of snow tomorrow.
So, chance of snowfall tomorrow= 40% = 0.4
The chances it will rain two days in a row = [tex]0.4 \times 0.3[/tex]
= [tex]0.12[/tex]
So, percent it will rain two days in a row = [tex]0.12 \times 100 = 12\%[/tex]
Hence the chances it will rain two days in a row is 12%
Employment data at a large company reveal that 74% of the workers are married, 42% are college graduates, and that 56% are married, given that they are college graduates. Which of the following statements are true about the events married and college graduate?
(A) These events are pairwise disjoint.
(B) These events are independent events.
(C) These events are both independent and pairwise disjoint.
(D) A worker is either married or a college graduate always.
(E) None of these above are true.
Answer:
(E) None of these above are true.
Step-by-step explanation:
Married = 74% or 0.74
College graduates = 42% or 0.42
pr(married | college graduates) = 0.56
(A) These events are pairwise disjoint. This is false. Pairwise disjoint are also known as mutually exclusive events. Here we can see that both events are occurring at same time.
(B) These events are independent events. This is also false.
(C) These events are both independent and pairwise disjoint. False
(D) A worker is either married or a college graduate always. False
Here Probability(A or B) shall be 1
= Pr(A) + Pr(B) - Pr( A and B) = 0.74 + 0.42 - 0.56 * 0.42 = 0.9248
This is not equal to 1.
(E) None of these above are true. This is true.
Jay said that by the quotient of powers property zero to the fifth power divided by zero to the second power equals zero to the fifth power -2 equals zero to the third power which equals zero is this correct explain
Answer:
No, Jay is not correct.
Step-by-step explanation:
Quotient of powers property:
For any non-zero number a and any integer x and y:
[tex]\frac{a^x}{a^y}=a^{x-y}[/tex]
According to by the quotient of powers property
[tex]\frac{0^5}{0^2}=0^{5-2}\Rightarrow 0^3=0[/tex]
We need to check whether Jay is correct or not.
No, Jay is not correct because quotient of powers property is used for non-zero numbers.
[tex]\frac{0^m}{0^n}=\frac{0}{0}=unde fined[/tex]
Therefore, Jay is not correct.
The charge to rent a trailer is $15 for up to 2 hours plus $8 per additional hour or portion of an hour. Find the cost to rent a trailer for 2.8 hours, 3 hours, and 8.5 hours. Then graph al ordered pairs, (hours, cost), for the function a. What is the cost to rent a trailer for 2.8 hours?
Answer:
The cost to rent a trailer for 2.8 hours is $21.4.
The cost to rent a trailer for 3 hours is $23.
The cost to rent a trailer for 8.5 hours is $67.
Step-by-step explanation:
Let x be the number of hours.
It is given that the charge to rent a trailer is $15 for up to 2 hours plus $8 per additional hour or portion of an hour.
The cost to rent a trailer for x hours is defined as
[tex]C(x)=\begin{cases}15 & \text{ if } x\leq 2 \\ 15+8(x-2) & \text{ if } x>2 \end{cases}[/tex]
For x>2, the cost function is
[tex]C(x)=15+8(x-2)[/tex]
We need to find the cost to rent a trailer for 2.8 hours, 3 hours, and 8.5 hours.
Substitute x=2.8 in the above function.
[tex]C(2.8)=15+8(2.8-2)=15+8(0.8)=21.4[/tex]
The cost to rent a trailer for 2.8 hours is $21.4.
Substitute x=3 in the above function.
[tex]C(3)=15+8(3-2)=15+8(1)=23[/tex]
The cost to rent a trailer for 3 hours is $23.
Substitute x=8.5 in the above function.
[tex]C(8.5)=15+8(8.5-2)=15+8(6.5)=67[/tex]
The cost to rent a trailer for 8.5 hours is $67.
Written all the ordered pairs in the form of (hours, cost).
(2.8,21.4), (3,23) and (8.5,67)
Plot these points on coordinate plane.
Final answer:
To find the cost to rent a trailer for 2.8 hours, we consider the flat fee of $15 for the first 2 hours and add the additional cost of $8 for the partial hour beyond 2 hours, resulting in a total cost of $23.
Explanation:
The cost to rent a trailer for a given number of hours is determined by a flat fee of $15 for the first 2 hours and an additional cost of $8 for each extra hour or partial hour. For 2.8 hours, since this exceeds the initial 2-hour period, we calculate the cost as follows:
Flat fee for the first 2 hours: $15Extra time: 0.8 hours, which counts as an additional full hour for charging purposesAdditional cost for 0.8 hours (rounded up to 1 hour): $8Total cost = Flat fee + cost for additional hoursTotal cost = $15 + $8 = $23Therefore, the cost to rent a trailer for 2.8 hours is $23.
Jordan received 9 text messages last week.She received 3 times more text message this week than last week. How many text messages did Jordan received this week?
Final answer:
Jordan received 27 text messages this week, which is 3 times more than the 9 messages she got last week.
Explanation:
Jordan received 9 text messages last week and this week she received 3 times more. To find out how many text messages Jordan received this week, multiply the number of messages from last week by 3.
9 text messages (last week) × 3 = 27 text messages (this week)
Therefore, Jordan received 27 text messages this week.
Jordan received 27 text messages this week. The average number of texts received per hour by a user is approximately 1.7292, based on the daily average. Precise probabilities for receiving exact or more than two messages per hour cannot be calculated without additional data.
Explanation:Jordan received 9 text messages last week and three times more this week. To calculate how many text messages Jordan received this week, we multiply 9 by 3, which is 9 × 3 = 27 text messages this week.
Solution for the Average Texts Received Per HourTo calculate the average texts received per hour, we divide the daily average by the number of hours in a day:
41.5 texts / 24 hours ≈ 1.7292 texts per hour.
Probability of Receiving Text Messagesa. The probability that a text message user receives or sends exactly two messages per hour is not provided in the given information and would typically require more data to calculate, like the distribution type. However, we know the average is 1.7292, so two messages is a little above average.
b. The probability of receiving more than two messages per hour involves determining the proportion of time users receive more than two messages, based on the average rate. Again, more information is needed to provide a precise probability.
Find the values of x1 and x2 where the following two constraints intersect.
1) 9x1 + 7x2 ≥ 57
(2) 4x1 + 6x2 ≥ 13
Constraints are simply the subjects of an objective function.
The point of intersection is: [tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]
The constraints are given as:
[tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]
[tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]
Express [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex] as an equation
[tex]\mathbf{4x_1 + 6x_2= 13}[/tex]
Subtract 6x2 from both sides
[tex]\mathbf{4x_1 = 13 - 6x_2}[/tex]
Divide through by 4
[tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)}[/tex]
Substitute [tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)} \\[/tex] in [tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]
[tex]\mathbf{9 \times \frac{1}{4}(13 - 6x_2) + 7x_2 \ge 57}[/tex]
Open brackets
[tex]\mathbf{29.25 - 13.5x_2 + 7x_2 \ge 57}[/tex]
[tex]\mathbf{29.25-6.5x_2 \ge 57}[/tex]
Collect like terms
[tex]\mathbf{-6.5x_2 \ge 57 - 29.25}[/tex]
[tex]\mathbf{-6.5x_2 \ge 27.25}[/tex]
Divide both sides by -6.5
[tex]\mathbf{x_2 \ge -4.19}[/tex]
Substitute -4.19 for x2 in [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]
[tex]\mathbf{4x_1 + 6 \times -4.19 \ge 13}[/tex]
[tex]\mathbf{4x_1 - 25.14 \ge 13}[/tex]
Add 25.14 to both sides
[tex]\mathbf{4x_1 \ge 38.14}[/tex]
Divide both sides by 4
[tex]\mathbf{x_1 \ge 9.54}[/tex]
Hence, the values are:
[tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]
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The solution of the problem involves finding the values of x1 and x2 which satisfy both inequalities when plotted on a graph. This can be done by simplifying the equations and comparing them.
Explanation:To solve this problem, we need to find where the two inequalities intersect. This means that we need to find the values of x1 and x2 which satisfy both inequalities.
Let's start with the first inequality '9x1 + 7x2 ≥ 57'. This means that the sum of 9 times x1 and 7 times x2 should be greater than or equal to 57. You can simplify this inequality by dividing the entire expression by the smallest coefficient which is 9, getting 'x1 + (7/9)x2 ≥ 57/9'.
Similarly, simplifying the second inequality '4x1 + 6x2 ≥ 13' by dividing by the smallest coefficient which is 4, we get 'x1 + (3/2)x2 ≥ 13/4'.
By comparing these two simplified inequalities, you should be able to identify the values of x1 and x2 where both inequalities are satisfied.
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what is the value of x if the average of 10, 11, 12 and x is 9?
Answer: The value of x is 3.
Step-by-step explanation:
Since we have given that
10,11, 12 and x
Average of above numbers = 9
As we know that
Average is given by
[tex]\dfrac{\text{Sum of observation}}{\text{Number of observation}}\\\\\\\dfrac{10+11+12+x}{4}=9\\\\10+11+12+x=9\times 4\\\\33+x=36\\\\x=36-33\\\\x=3[/tex]
Hence, the value of x is 3.
Producers will make 1000 refrigerators available when the unit price is $280.00. At a unit price of $400, 5000 refrigerators will be marketed . Find the equation relating the unit price p of a refrigerator to the quantity supplied x if the equation is known to be linear.
p =
How many refrigerators will be marketed when the unit price is $440?______ refrigerators
What is the lowest price at which a refrigerator will be marketed? $_________ (lowest price)
Answer:
For linear equations we use:
p=mx+b ------ (1)
Now we have the following coordinates:
(x1,p1)= (1000,280) and (x2, p2)=(5000,400)
First we need slope (m)
m= [tex](400-280)/(5000-1000)[/tex]
= [tex]120/4000=0.03[/tex]
Now we will plug the value of m in the first equation
[tex]280=0.03(1000)+b[/tex]
=> [tex]280=30+b[/tex]
=> b = 250
Now plug into p=mx+b using only m=0.03 and b=250
[tex]p=0.03x+250[/tex]
When the unit price is $440, we can plug in 440 in for p;
[tex]440=0.03x+250[/tex]
=> [tex]0.03x=440-250[/tex]
=> [tex]0.03x=190[/tex]
=> x = 6333 refrigerators
The lowest price at which a refrigerator will be marketed, we can find this by plugging x = 0 in p=mx+b.
[tex]p=0.03(0)+250[/tex]
=> p = $250
The linear equation relating the unit price p to the quantity supplied x is p = 0.03x + 250. When the unit price is $440, approximately 6333 refrigerators will be marketed. The lowest price at which a refrigerator will be marketed is $250.
To find the equation relating the unit price p of a refrigerator to the quantity supplied x when the relationship is known to be linear, we can use the two given points: (1000, 280) and (5000, 400).
First, we determine the slope (m) of the line:
m = (400 - 280) / (5000 - 1000) = 120 / 4000 = 0.03
Next, we use the point-slope form of the equation y - y₁ = m(x - x₁) where (x₁, y₁) is one of our points. We can use (1000, 280):
p - 280 = 0.03(x - 1000)
p = 0.03x + 250
Now, let's determine how many refrigerators will be marketed when the unit price is $440:
440 = 0.03x + 250
190 = 0.03x
x = 6333.33
So, approximately 6333 refrigerators will be marketed when the unit price is $440.
Lastly, we find the lowest price at which a refrigerator will be marketed by setting x to 0:
p = 0.03(0) + 250 = 250
The lowest price at which a refrigerator will be marketed is $250.
The u.s. senate consists of 100 members, 2 from each state. A committee of five senators is formed. What is probability it contains one senator from your state?
Answer:
The probability is 9.80%.
Step-by-step explanation:
The u.s. senate consists of 100 members, 2 from each state.
A committee of five senators is formed.
P(at least one from Your state) = 1- [tex]\frac{98c5}{100c5}[/tex]
= 1- [tex]\frac{67910864}{75287520}[/tex]
= [tex]1-0.9020[/tex]
= 0.098
That is, 9.80%.
in the following ordinary annuity, the interest is compounded with each payment, and the payment made at the end of the compounding period. find the accumulated amount of the annuity. (round your answer to the nearest cent)
$2000 monthy at 6.3% for 20 years
Answer: $957646.07
Step-by-step explanation:
The formula we use to find the accumulated amount of the annuity is given by :-
[tex]FV=m(\frac{(1+\frac{r}{n})^{nt})-1}{\frac{r}{n}})[/tex]
, where m is the annuity payment deposit, r is annual interest rate , t is time in years and n is number of periods.
Given : m= $2000 ; n= 12 [∵12 in a year] ; t= 20 years ; r= 0.063
Now substitute all these value in the formula , we get
[tex]FV=(2000)(\frac{(1+\frac{0.063}{12})^{12\times20})-1}{\frac{0.063}{12}})[/tex]
i.e. [tex]FV=(2000)(\frac{(1+0.00525)^{240})-1}{0.00525})[/tex]
i.e. [tex]FV=(2000)(\frac{(3.51382093497)-1}{0.00525})[/tex]
i.e. [tex]FV=(2000)(\frac{2.51382093497}{0.00525})[/tex]
i.e. [tex]FV=(2000)(478.823035232)[/tex]
i.e. [tex]FV=957646.070464\approx957646.07\ \ \ \text{ [Rounded to the nearest cent]}[/tex]
Hence, the accumulated amount of the annuity= $957646.07
The future value or accumulated amount of an ordinary annuity is calculated using the formula where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years. Given P = $2000, r = 6.3%, n = 12 and t = 20 years, substituting these values into the formula gives the accumulated amount
Explanation:To find the future value or accumulated amount of an ordinary annuity, we use the formula: FV = P * (((1 + r)^nt - 1) / r), where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years.
In the given problem, P = $2000, r = 6.3% or 0.063 (in decimal), n = 12 (since the payments are monthly), and t = 20 years.
Substituting these into the formula, FV = $2000 * (((1 + 0.063 /12)^(12*20) - 1) / (0.063/12)).
Calculating the equation, we'll get the accumulated amount to the nearest cent.
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An airplane left airport A flying on a course of 72 degrees.
After flying 495 km, the plane was just able to pick up
communication signals from airport B which is due east of airport
A. If airport B's communication signals can be picked up within a
radius of 300 km from airport B, for how many kilometers can the
airplane fly and still be in contact with airport B? Give you
answer to one decimal place.
Distance for which Aeroplane can be in contact with Airport B is = 396.34 km
Step-by-step explanation:In the question,
We have an Airport at point A and another at point B.
Now,
Airplane flying at the angle of 72° with vertical catches signals from point D.
Distance travelled by Airplane, AD = 495 km
Now, Let us say,
AB = x
So,
In triangle ABD, Using Cosine Rule, we get,
[tex]cos(90-72) =cos18= \frac{AB^{2}+AD^{2}-BD^{2}}{2.AD.AB}[/tex]
So,
On putting the values, we get,
[tex]cos18 = \frac{x^{2}+495^{2}-300^{2}}{2(495)(x)}\\0.951(990x)=x^{2}+245025-90000\\x^{2}-941.54x+155025=0\\[/tex]
Therefore, x is given by,
x = 212.696, 728.844
So,
The value of x can not be 212.696 as the length of LB (radius) itself is 300 km.
So,
x = 728.844 km
So,
AL = AB - BL
AL = x - 300
AL = 728.844 - 300
AL = 428.844 km
Now, in the circle from a property of secants we can say that,
AL x AM = AD x AC
So,
428.844 x (728.844 + 300) = 495 x AC
441213.576 = 495 x AC
AC = 891.34 km
So,
The value of CD is given by,
CD = AC - AD
CD = 891.34 - 495
CD = 396.34 km
Therefore, the distance for which the Aeroplane can still be in the contact with Airport B is 396.34 km.
timy earns $7.75 from his part-time job at walmart. He worked 5 hours on monday, 3 hours on wednesday and 7 hours on friday. How much money did timmy earn this week?
Answer:
Timy earned $116.25 this week.
Step-by-step explanation:
Timy earns $7.75 from his part-time job at Walmart.
Let us suppose this is his hourly rate.
He worked 5 hours on Monday, 3 hours on Wednesday and 7 hours on Friday.
So, total hours he worked = [tex]5+3+7=15[/tex] hours
Now, his earnings will be = [tex]15\times7.75=116.25[/tex] dollars
Therefore, Timy earned $116.25 this week.
Answer:
$116.25
Step-by-step explanation:
Timy earns $7.75 from his part-time job at Walmart.
He worked on Monday = 5 hours
On Wednesday = 3 hours
On Friday = 7 hours
Total earning of this week = (7.75 × 5) + (7.75 × 3) + (7.75 × 7)
= 38.75 + 23.25 + 54.25
= $116.25
Timy earned $116.25 this week.
there are 3 times as many boys as girls. If there are 24 more boys than girls, how many children are there?
How would I put this into a bar diagram? I can do it with algebra but I am having trouble drawing a bar diagram to help represent this problem.
Answer:
There are 48 children.
Step-by-step explanation:
Given :There are 3 times as many boys as girls.
There are 24 more boys than girls,
To Find : how many children are there?
Solution:
Let the number of girls be x
Now we are given that there are 3 times as many boys as girls.
So, no. of boys = 3x
Now we are given that there are 24 more boys than girls.
So, [tex]3x-x=24[/tex]
[tex]2x=24[/tex]
[tex]x=12[/tex]
So, no. of girls = 12
No. of boys = 3x = 3(12) = 36
Now the total no. of children = 12+36 = 48
Hence there are 48 children.
Sprint Read the problems below, and record your answers in a Word document. Be sure to show your work! 4:15 PM . Questions 1 and 2: You must prepare 400mL of a solution that requires a 1:8 concentration of drug. Sterile water is the diluent you should use How much drug do you need for this medication? How much diluent do you need?
Answer:
The amount of drug required = 44.44 mL
Diluent needed = 355.56 mL
Step-by-step explanation:
Data provided in the question:
Total volume of solution = 400 mL
Concentration of drug = 1 : 8
Now,
The ratio is interpreted as 1 part of drug and 8 part of diluent
Thus,
The amount of drug required = [tex]\frac{1}{1+8}\times\textup{Total volume of solution}[/tex]
or
The amount of drug required = [tex]\frac{1}{1+8}\times\textup{400 mL}[/tex]
or
The amount of drug required = 44.44 mL
and,
Diluent needed = [tex]\frac{8}{1+8}\times\textup{400 mL}[/tex]
or
Diluent needed = 355.56 mL
Final answer:
To make a 400mL solution with a 1:8 drug concentration, you need 44.4mL of the drug and 355.6mL of sterile water.
Explanation:
To prepare 400mL of a solution with a 1:8 concentration of a drug, using sterile water as the diluent, we should first calculate the amount of drug needed. A 1:8 concentration ratio means that for every 1 part drug, there are 8 parts diluent. Therefore, the total number of parts is 1 (drug) + 8 (diluent) = 9 parts.
To find the amount of drug needed:
Divide the total volume of the solution by the total number of parts:To find the amount of diluent needed:
Subtract the volume of the drug from the total volume of the solution:To summarize, you need 44.4mL of the drug and 355.6mL of sterile water to make a 400mL solution with a 1:8 drug concentration.
solve showing work for absolute value
solve problem show work
Steven just receiveda 3% raise. His new salary is $55,620.
What was his old salary?
Appliance store hwere ramios shops offers a 9% discount for
paying cah. If the price of a new frige without a discout is $978,
find the discount and the new price.
A 6 ft board is cut into 2 peices so that one peice is 2 feet
longer than 3 times the shorter peice. Ifthe shorter peice is x
fett long, find the lengths of both peices.
Answer:
$54,000
Amount of discount = $88.02
The new price = $978 - $88.02 = $889.98
Length of shorter piece is 1 ft and longer piece is 5 ft
Step-by-step explanation:
Given:
Raise received = 3%
New salary = $55,620
Now,
New salary = old salary + 3% of old salary
or
$55,620 = old salary + (0.03 × old salary)
or
$55,620 = Old salary × (1.03)
or
Old salary = $54,000
Given:
Price of the new fridge = $978
Discount offered = 9%
Thus,
Amount of discount = 9% of $978
or
Amount of discount = 0.09 × $978
or
Amount of discount = $88.02
And, the new price = Price of the fridge - Amount of discount
or
The new price = $978 - $88.02 = $889.98
Given:
Length of the of the board before cutting = 6 ft
Now,
According to the question
let the length of the shorter piece be 'x'
thus,
6 = x + (3x + 2)
or
6 = 4x + 2
or
4 = 4x
or
x = 1 ft
hence,
shorter piece is 1 ft long and longer piece is (3x +2 = 5ft)
Sixty-seven percent of adults have looked at their credit score in the past six months. If you select 31 customers, what is the probability that at least 20 of them have looked at their score in the past six months? 0.450 0.550 0.692 0.142
Answer:
0.692.
Step-by-step explanation:
This is a Binomial Probability of Distribution with P(success) = 0.67. Prob success >= 20) , 31 trials.
From Binomial Tables we see that the required probability = 0.692.
The cost, in dollars, of making x items is given by the function C(x)=25x+300
. Find C(1500), the cost of making 1500 items.
Question 6 options:
a)
$4050
b)
$25,300
c)
$30,300
d)
$37,800
e)
none
Answer:
d) $37800
Step-by-step explanation:
Cost of making x items = [tex]C(x)=25x + 300[/tex]
Cost of making [tex]1500[/tex] items = [tex]C(1500)=25(1500) + 300\\C(1500)= 37500 + 300\\C(1500)= 37800[/tex]
Cost of making [tex]1500[/tex] items = $37800
d) $37800 is the correct answer
d) Find the 5th degree Taylor polynomial centered at x = 0 for the function y = \tiny \frac{x}{1+x}
Answer:
[tex]P(x) = x - x^2 + x^3 - x^4+x^5[/tex]
Step-by-step explanation:
Let us first remember how a Taylor polynomial looks like:
Given a differentiable function [tex]f[/tex] then we can find its Taylor series to the [tex]nth[/tex] degree as follows:
[tex]P(x) = f(x_{0}) + f'(x_{0}).(x-x_{0}) + \frac{f''(x_{0})}{2!}.(x-x_{0})^2+.....+\frac{f^n(x_{0})}{n!}.(x-x_{0})^n + R_{n}(x).(x-x_{0})^n[/tex]
Where [tex]R_{n}(x)[/tex] represents the Remainder and [tex]f^n(x)[/tex] is the [tex]nth[/tex] derivative of [tex]f[/tex].
So let us find those derivatives.
[tex]f(x) = \frac{x}{1+x}\\f'(x) = \frac{1}{(1+x)^2}\\f''(x) = \frac{-2}{(1+x)^3}\\f'''(x) = \frac{6}{(1+x)^4}\\f''''(x) = \frac{-24}{(1+x)^5}\\f'''''(x) = \frac{120}{(1+x)^6}[/tex]
The only trick for this derivatives is for the very first one:
[tex]f'(x) = \frac{1}{1+x} - \frac{x}{(1+x)^2}\\f'(x) = \frac{(1+x) - x}{(1+x)^2} = \frac{1}{(1+x)^2}\\[/tex]
Then it's only matter of replacing on the Taylor Series and replacing [tex]x_{0}=0[/tex]
The marketing team at Beth's Butter Works decided they preferred the traditional plastic tub packaging, but they wanted a more refined estimate of potential sales. They launched a third test at a regional level across 100 stores. These 100 stores had average daily sales of 140 units with a standard deviation of 50.
Calculate the 99%, 95%, and 68% confidence intervals for the average number of units Beth's Butter Works can anticipate to sell. Enter your results in the blanks below.
There is a 99% likelihood that they will sell between ___ and ___ units.
There is a 95% likelihood that they will sell between ___ and ___ units.
There is a 68% likelihood that they will sell between ___ and ___ units.
There is a 99% likelihood that they will sell between 127.12 and 152.88 units.
There is a 95% likelihood that they will sell between 130.2 and 149.8 units.
There is a 68% likelihood that they will sell between 135 and 145 units.
Use the concept of the confidence interval of statistics defined as:
In statistics, a confidence interval describes the likelihood that a population parameter would fall between a set of values for a given percentage of the time. Confidence ranges that include 95% or 99% of anticipated observations are frequently used by analysts.
Given that,
The marketing team at Beth's Butter Works prefers the traditional plastic tub packaging.
They wanted a more refined estimate of potential sales.
They launched a third test at a regional level across 100 stores.
The average daily sales of these 100 stores during the test period was 140 units.
The standard deviation of daily sales across the 100 stores was 50 units.
To calculate the confidence intervals:
Consider the sample mean, sample standard deviation, and the desired level of confidence.
In this case,
Use the average daily sales of 140 units and the standard deviation of 50 units.
Now,
For a 99% confidence interval:
Use a z-score of 2.576 (corresponding to a 99% confidence level).
The formula for the confidence interval is:
Confidence Interval[tex]=\text{ Sample Mean} \pm (\text{Z-Score} \times (\text{Sample Standard Deviation} /\sqrt{\text{Sample Size}}))[/tex]
For a 99% confidence interval, the values are:
Lower bound [tex]= 140 - (2.576 \times (50 / \sqrt{100}))[/tex]
Lower bound = 127.12
Upper bound [tex]= 140 + (2.576 \times (50 / \sqrt{100}))[/tex]
Upper bound = 152.88
For a 95% confidence interval:
Use a z-score of 1.96 (corresponding to a 95% confidence level).
The values are:
Lower bound = [tex]140 - (1.96 \times (50 / \sqrt{100}))[/tex]
Lower bound = 130.2
Upper bound = [tex]140 + (1.96 \times (50 / \sqrt{100}))[/tex]
Upper bound = 149.8
For a 68% confidence interval:
Use a z-score of 1 (corresponding to a 68% confidence level).
The values would be:
Lower bound [tex]= 140 - (1 \times(50 / \sqrt{100}))[/tex]
Lower bound = 135
Upper bound [tex]= 140 + (1 \times(50 / \sqrt{100}))[/tex]
Upper bound = 145
Hence,
99% confidence interval:
There is a 99% likelihood that they will sell between 127.12 and 152.88 units.
95% confidence interval:
There is a 95% likelihood that they will sell between 130.2 and 149.8 units.
68% confidence interval:
There is a 68% likelihood that they will sell between 135 and 145 units.
To learn more about statistics visit:
https://brainly.com/question/30765535
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Final answer:
To calculate the confidence intervals for the average number of units sold by Beth's Butter Works, the formula for confidence intervals is used with a mean of 140 units and a standard deviation of 50 across 100 stores. The 99%, 95%, and 68% confidence levels correspond to confidence intervals of 127.12 to 152.88 units, 130.2 to 149.8 units, and 135 to 145 units, respectively.
Explanation:
To calculate the confidence intervals for the average number of units sold, we will use the formula for a confidence interval: mean ± (z * (standard deviation / √(sample size))). The mean daily sales are 140 units and the standard deviation is 50. Since the sample size is 100 stores, the standard error (standard deviation / √(sample size)) would be 50 / √(100) = 50 / 10 = 5. The z-scores for the different confidence levels are approximately 2.576 for 99%, 1.96 for 95%, and 1 for 68% (as this lies closest to one standard deviation from the mean).
For a 99% confidence interval, the calculation is:
140 ± (2.576 * 5) = 140 ± 12.88
The 99% confidence interval is therefore between 127.12 and 152.88 units.
For a 95% confidence interval, the calculation is:
140 ± (1.96 * 5) = 140 ± 9.8
The 95% confidence interval is therefore between 130.2 and 149.8 units.
To explain what a 95% confidence interval means for this study, it implies that, if we were to take many samples and build confidence intervals in the same way, 95% of them would contain the true average units sold across all possible stores.
For a 68% confidence interval, the calculation is:
140 ± (1 * 5) = 140 ± 5
The 68% confidence interval is therefore between 135 and 145 units.
Given any set of 53 integers, show that there are two of them having the property that either their sum or their difference is evenly divisible by 103. (This is a Pigeonhole principle problem).
Answer:
See proof below.
Step-by-step explanation:
The Pigeonhole principle states that if we place n+1 objects in n places, then one of those n places must have more than one object. In theory, this may seem a very obvious principle but some of the problems which involve this principle can be more difficult than what you'd think of.
In this case we have to prove that given ANY set of integers, there are two of them having the property that either their sum or their difference is evenly divisible by 103.
This would translate to: if we have n and m integers in this set, we'd have one pair for which 103|(n+m) or 103|(n-m). This last condition gives us the clue of using modulos for this problem.
First, we're going to choose 52 pigeonholes (since we have 53 integers). Now, we're going to label the integers with numbers from 0 to 102 depending on their congruence modulo 103.
Once we've done this, we're going to place the integers in the pigeonhole according to their congruence, the pigeonholes will be numbered (0,103), (1,102), (2,101), (3,100)... (50,53), (51,52). (I.e: If the integer is congruent to 6 modulo 103, it will be placed in the (6,97) pigeonhole).
This way any two integers that are placed in one of these pigeonholes will be divisible by 103 (either their sum or their difference).
Note that we have 52 pigeonholes and 53 integers, therefore, one of the pigeonholes will have more than one number (two at least) and that's how we are sure it will satisfy the relation that their sum or their difference is evenly divisible by 103.
What is the lenght of a diagonal of a square with sides 16feet
long ? Round to the nearest tenth.
Answer:
The length of the diagonal is 22.6 ft.
Step-by-step explanation:
To find the length of the diagonal of a square, multiply the length of one side by the square root of 2:
If the length of one side is x, [tex]length = x\sqrt{2}[/tex] as you can see in the image attached.
This fact is a consequence of applying the Pythagoras' Theorem to find the length of the diagonal if we know the side length of the square.
[tex]length^{2} = x^{2}+x^{2} \\ length=\sqrt{x^{2}+x^{2}} \\ length=\sqrt{2x^{2} } \\ length=x\sqrt{2}[/tex]
We know that the length of one side is 16 ft so [tex]length = 16\sqrt{2}=22.627[/tex] and round to the nearest tenth is 22.6 ft
In experiment two, I measured a block to be 4.45 cm by 3.35 em by 6.15 cm and a mass of 155.147g. What is the density of this block in kg/L? 3.
Answer:
1.69224581396 Kg/L
Step-by-step explanation:
We are given the measure of the block as 4.45 cm × 3.35 cm × 6.15 cm.
Volume of block = 4.45 cm × 3.35 cm × 6.15 cm = 91.681125 cm cube = 91.681125 × 0.001 L = 0.091681125 L
We did the above step to convert the volume of block into Liter.
Mass of block is given as 155.147 gram = 155.147 × 0.001 kg = 0.155147 kg
We converted the mass of block into kilograms because we need density in Kg/L.
Density is defined as mass per unit volume
Density = [tex]\frac{Mass}{Volume}[/tex]
= [tex]\frac{0.155147 }{0.091681125}[/tex]]
= 1.69224581396 Kg/L
The density is found to be approximately 1.688 kg/L.
To find the density of the block, we need to use the density formula:
Density = Mass / Volume
The given dimensions of the block are:
Length = 4.45 cmWidth = 3.35 cmHeight = 6.15 cmFirst, calculate the volume:
Volume = Length × Width × Height
Volume = 4.45 cm × 3.35 cm × 6.15 cm
Volume ≈ 91.88925 cubic centimeters (cm)
Next, convert mass to kilograms and volume to liters:
Mass = 155.147 g = 0.155147 kg (since 1 kg = 1000 g)Volume = 91.88925 cm³ × 0.001 L/cm³ ≈ 0.09188925 L (since 1 cm³ = 0.001 L)Finally, calculate the density in kg/L:
Density = Mass / Volume
Density ≈ 0.155147 kg / 0.09188925 L
Density ≈ 1.688 kg/L
Thus, the density of the block is approximately 1.688 kg/L.
Graph the equation by first finding the x- andy-intercepts.
one half x plus one third y equals 1
Answer:
x-intercept: (2,0)
y-intercept: (0,3)
Step-by-step explanation:
We are asked to graph our given equation [tex]\frac{1}{2}x+\frac{1}{3}y=1[/tex].
To find x-intercept, we will substitute [tex]y=0[/tex] in our given equation.
[tex]\frac{1}{2}x+\frac{1}{3}(0)=1[/tex]
[tex]\frac{1}{2}x+0=1[/tex]
[tex]2*\frac{1}{2}x=2*1[/tex]
[tex]x=2[/tex]
Therefore, the x-intercept is [tex](2,0)[/tex].
To find y-intercept, we will substitute [tex]x=0[/tex] in our given equation.
[tex]\frac{1}{2}(0)+\frac{1}{3}y=1[/tex]
[tex]0+\frac{1}{3}y=1[/tex]
[tex]3*\frac{1}{3}y=3*1[/tex]
[tex]y=3[/tex]
Therefore, the y-intercept is [tex](0,3)[/tex].
Upon connecting these two points, we will get our required graph as shown below.
In a particular game, a fair die is tossed. If the number of spots showing is six you win $6, if the number of spots showing is five you win $3, and if the number of spots showing is four you win $1. If the number of spots showing is one, two, or three you win nothing. You are going to play the game twice. Each game is independent. The probability that you will win nothing on the two plays of the game is
Answer: Our required probability is 0.194.
Step-by-step explanation:
Since we have given that
Amount win for showing 6 = $6
Amount win for showing 5 = $3
Amount win for showing 4 = $1
Amount win for showing 1, 2, 3 = $0
So,we need to find the probability that he will win nothing on the two plays of the game.
so, the outcomes would be
(1,1), (1,2), (1,3), (2,1), (3,1),(2,2), (3,3)
So, Number of outcomes = 7
total number of outcomes = 36
So, Probability of wining nothing = [tex]\dfrac{7}{36}=0.194[/tex]
Hence, our required probability is 0.194.
Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units, two-bedroom townhouses, and three-bedroom townhouses. A total of 216 units is planned. The total number of two- and three-bedroom townhouses will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of three-bedroom townhouses, find how many units of each type will be in the complex. one-bedroom units units two-bedroom townhouses units three-bedroom townhouses units
Answer:
108 one-bedroom units
72 two-bedroom units
36 three-bedroom units
Step-by-step explanation:
Let x, y, z the number of one-bedroom, two-bedroom and three-bedroom units respectively. Then
1) x+y+z = 216
2) y+z = x
3) x = 3z
Multiplying equation 1) by -1 and adding it to 2), we get
-x = x-216 so, x = 216/2 = 108
x = 108
Replacing this value in 3) we get
z = 108/3 = 36
z = 36
Replacing now in 2)
y+36 = 108, y = 108-36 and
y = 72
In the planned apartment complex, there will be 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses.
Explanation:Let x be the number of one-bedroom units. Since the number of two- and three-bedroom townhouses equals the number of one-bedroom units, let y be the number of two-bedroom townhouses and z be the number of three-bedroom townhouses. We know that x + y + z = 216. Additionally, x = 3z because the number of one-bedroom units will be 3 times the number of three-bedroom townhouses. Substituting x = 3z into the first equation gives 3z + y + z = 216. Simplifying this equation, we get 4z + y = 216.
Now, we can solve this system of equations to find the values of x, y, and z. Subtracting y from both sides of the equation 4z + y = 216 gives 4z = 216 - y. Let's call this equation (1). Substituting x = 3z and y = 216 - 4z into the equation x + y + z = 216 gives 3z + (216 - 4z) + z = 216. Simplifying this equation, we get 4z + 216 = 216. Subtracting 216 from both sides of the equation gives 4z = 0. Let's call this equation (2).
Since equation (1) and equation (2) both have 4z on the left side, we can equate the right sides of the equations. This gives 216 - y = 0. Solving for y, we find y = 216. Plugging this value of y into equation (1), we get 4z = 216 - 216, which simplifies to 4z = 0. Solving for z, we find z = 0. Finally, plugging the value of z into the equation x = 3z, we get x = 3(0), which simplifies to x = 0.
Therefore, there are 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses in the complex.