Answer:
The correct answer here is option c) 0.05 mg(milligram)
Step-by-step explanation:
The first step here to calculate the milligram of active ingredient in a 5% solution per milliliter is to assume that Z milligram of active ingredient is present in the 5% solution per milliliter.
NOTE* - milligram ( mg) and milliliter (ml)
so therefore, the equation would become,
5\% = \frac{Z mg}{1 ml}
.05 = \frac{Z mg}{1 ml}
.05 x 1 = Z mg
therefore there is .05 mg of active ingredient in the 5% solution per ml.
The correct option is A. 50 mg. This is calculated by knowing that 5% means 5 grams or 5000 milligrams in 100 millilitres, which divides down to 50 milligrams per millilitre.
To determine how many milligrams of active ingredients are in a 5% solution per millilitres, we need to understand what a 5% solution means. A 5% solution means that there are 5 grams of the active ingredient in every 100 millilitres of the solution.
Let's break it down step-by-step:
→ 5% solution means 5 grams of active ingredient per 100 millilitres (mL) of solution.
→ Convert grams to milligrams: 5 grams = 5000 milligrams.
→ Therefore, 100 mL of solution contains 5000 milligrams of active ingredient.
To find out how many milligrams are in 1 mL, we divide the total milligrams by the total millilitres:
→ 5000 milligrams / 100 mL = 50 milligrams per mL
Thus, the correct answer is A. 50 mg.
Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?
The fourth point that will complete the quadrilateral is a rhombus is -3.
In a rhombus, two parallel sides must be equal.
What is a straight line graph?The graph follows a straight line equation shows a straight line graph.equation of a straight line is y=mx+cy represents vertical line y-axis.x represents the horizontal line x-axis. m is the slope of the lineslope(m)=tan∅=y axis/x axis.
c represents y-intercepts (it is the point at which the line cuts on the y-axis)Straight line graphs show a linear relationship between the x and y values.Calculation:-
1st coordinate- side=[tex]\sqrt{10}[/tex]
2nd coordinate- side= [tex]\sqrt{18}[/tex]
3rd coordinate must be [tex]\sqrt{10}[/tex]
so,y=-3
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Answer:
Consider the three points graphed. What is the value of y in the fourth point that will complete the quadrilateral as a rhombus?
(–1, -5 )
Step-by-step explanation:
An item is discounted 20%; the sale price after the discount is $60. What was the original price? Round your answer to the nearest two decimal digits and express your answer without the $ sign (e.g., 1234.25, not $1234.25)
Answer:
75.
Step-by-step explanation:
Let x be the original price. 20% = 0.2. The price with the discount is 60, that is
x-x*0.2 = 60
x(1-0.2) = 60 using common factor,
x (0.8) = 60
x = 60/0.8
x = 75.
So, the original price is $75.
The original price of an item discounted 20% with a sale price after the discount of $60 is $75.
What is discounted price?A discounted price is the marked-down price of an item.
The discounted price represents the selling price after reducing it with the discount.
Data and Calculations:Discount rate = 20%
Discounted price = $60
Original price = $75 ($60/1-20%)
Thus, the original price of an item discounted 20% with a sale price after the discount of $60 is $75.
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Find and simplify each of the following for
Answer:
(A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
(B) f(x + h) - f(x) = 8xh + 4h² - 6h
(C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
Step-by-step explanation:
* Lets explain how to solve the problem
- The function f(x) = 4x² - 6x + 6
- To find f(x + h) substitute x in the function by (x + h)
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) = 4(x + h)² - 6(x + h) + 6
- Lets simplify 4(x + h)²
∵ (x + h)² = (x)(x) + 2(x)(h) + (h)(h) = x² + 2xh + h²
∴ 4(x + h)² = 4(x² + 2xh + h²) = 4x² + 8xh + 4h²
- Lets simplify 6(x + h)
∵ 6(x + h) = 6(x) + 6(h)
∴ 6(x + h) = 6x + 6h
∴ f(x + h) = 4x² + 8xh + 4h² - (6x + 6h) + 6
- Remember (-)(+) = (-)
∴ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
* (A) f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
- Lets find f(x + h) - f(x)
∵ f(x + h) = 4x² + 8xh + 4h² - 6x - 6h + 6
∵ f(x) = 4x² - 6x + 6
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - (4x² - 6x + 6)
- Remember (-)(-) = (+)
∴ f(x + h) - f(x) = 4x² + 8xh + 4h² - 6x - 6h + 6 - 4x² + 6x - 6
- Simplify by adding the like terms
∴ f(x + h) - f(x) = (4x² - 4x²) + 8xh + 4h² + (- 6x + 6x) - 6h + (6 - 6)
∴ f(x + h) - f(x) = 8xh + 4h² - 6h
* (B) f(x + h) - f(x) = 8xh + 4h² - 6h
- Lets find [tex]\frac{f(x+h)-f(x)}{h}[/tex]
∵ f(x + h) - f(x) = 8xh + 4h² - 6h
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh + 4h^{2}-6h}{h}[/tex]
- Simplify by separate the three terms
∴ [tex]\frac{f(x+h)-f(x)}{h}=\frac{8xh}{h}+\frac{4h^{2} }{h}-\frac{6h}{h}[/tex]
∴ [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
* (C) [tex]\frac{f(x+h)-f(x)}{h}=8x+4h-6[/tex]
The one-to-one functions g and h are defined as follows. g = {(24), (4, 2), (5,-1), (8, -9) h x) 8x+13 Find the following. g (4) =
Answer:
The value of g(4) is 2.
Step-by-step explanation:
Given,
The one-to-one functions g is defined as,
g = {(2,4), (4, 2), (5,-1), (8, -9)}
The element of a function is written in the form of order pair where, the first element is input value and second element is the output value.
Since, under the function g the output value of input value 2 is 4,
That is, we can write,
g(2) = 4
Hence, the value of g(4) is 2.
Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 131 millimeters, and a standard deviation of 7 millimeters. If a random sample of 31 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by more than 1.9 millimeters? Round your answer to four decimal places.
Answer: 0.1310
Step-by-step explanation:
Given : Mean : [tex]\mu = \text{131 millimeters}[/tex]
Standard deviation : [tex]\sigma = \text{7 millimeters}[/tex]
Sample size : [tex]n=31[/tex]
To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters i.e. less than 129.1 milliliters and less than 132.9 milliliters.
The formula for z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 129.1 milliliters
[tex]z=\dfrac{129.1-131}{\dfrac{7}{\sqrt{31}}}\approx-1.51[/tex]
For x = 132.9 milliliters
[tex]z=\dfrac{132.9-131}{\dfrac{7}{\sqrt{31}}}\approx1.51[/tex]
The P-value= [tex]P(x<-1.51)+P(x>1.51)[/tex]
[tex]=2P(z>1.51)=2(1-P(z<1.15))\\=2(1-0.9344783)\\=0.1310434\approx0.1310[/tex]
Hence, the required probability = 0.1310
The probability that the sample mean would differ from the population mean by more than 1.9 millimeters is approximately 0.1744.
Explanation:To find the probability that the sample mean would differ from the population mean by more than 1.9 millimeters, we can use the Z-score formula. The Z-score is calculated by subtracting the population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. Once we have the Z-score, we can use a Z-table or calculator to find the probability.
In this case, the Z-score is (1.9 - 0) / (7 / sqrt(31)) = 0.935. To find the probability of a Z-score greater than 0.935, we can look up the corresponding area under the normal distribution curve in a Z-table or use a calculator. The probability is approximately 0.1744.
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Assume that photo coordinates (in mm) of points a and b are Xa=65.35, Ya=74.88 and Xb=31.45, Yb= -55.50. What is the photo distance ab?
(a) 134.7
(b) 39.05
(c) 162.4
(d) 92.68
Answer:
Option A (134.7mm)
Step-by-step explanation:
Let's find the distance, but first we need to remember that the distance between two points with coordinates (Xa,Ya) and (Xb,Yb) is defined by:
[tex]distance = \sqrt{(Xb-Xa)^{2} + (Yb-Ya)^{2} }[/tex]
From the situation we notice that:
Xb=31.45 and Xa=65.35, as well as:
Yb=-55.50 and Ya=74.88
Using the previous equation we have:
[tex]distance = \sqrt{(31.45-65.35)^{2} + (-55.50-74.88)^{2} }[/tex]
[tex]distance = \sqrt{(-33.9)^{2} + (-130.38)^{2} }[/tex]
[tex]distance = \sqrt{1149.21 + 16998.9444}[/tex]
[tex]distance = \sqrt{18148.1544}[/tex]
[tex]distance = 134.7151mm[/tex]
In conclusion, the distance between points (65.35,74.88) and (31.45,-55.50) is 134.7151mm, which is option A (134.7mm).
Solve the given differential equation by using an appropriate substitution. The DE is of the form dy dx = f(Ax + By + C), which is given in (5) of Section 2.5. dy dx = 4 + y − 4x + 5
No idea what the cited section's method is, but this ODE is linear:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]
Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:
[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]
Integrating both sides gives
[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]
[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]
Prove: For all sets AB,and CIA-B) U (A n B) = A
Answer:
Step-by-step explanation:
(A-B)U (A∩B) [ We know that difference of A and B is A∩B')
A∩B')U(A∩B)
A∩(B'UB) (Using Distributive law)
A∩(U) [ Union of a set and its complement is always a universal set]
A Hence proved
How much would be in your savings account in eight years after depositing $180 today if the bank pays 8 percent per year? (Do not round intermediate calculations. Round your answer to 2 decimal places.) 10 points Future value Skipped eBook Hint Print Referer LO
Answer:
$333.17
Step-by-step explanation:
Use the compounding formula
[tex]A(t)=P(1+r)^t[/tex]
where A(t) is the amount at the end of the compounding,
P is the initial deposit,
r is the interest rate in decimal form, and
t is the time in years.
Filling in our info:
[tex]A(t)=180(1+.08)^8[/tex]
Simplify a bit to
[tex]A(t)=180(1.08)^8[/tex]
Raise 1.08 to the 8th power and get
A(t) = 180(1.85093021) and then multiply to get
A(t) = $333.17
Suppose that you follow a population over time. When you plot your data on a semilog plot (using logs with base 10), a straight line with slope 0.1 results. Furthermore, assume that the population size at time 0 was 80. What function best describes the population size at time t?
Answer:
[tex]P(t)=80(10^{0.1t})[/tex]
Step-by-step explanation:
The 'y' axis represent log(P), so it may be modeled as a line (or linear function), where its slope is 0.1:
[tex]log(P)=0.1t+C[/tex]
Pow each part of the equation by 10:
[tex]10^{log(P)}=10^{0.1t+C}\\ P=10^{0.1t+C}[/tex]
Evaluate at t=0, where the population is known.
[tex]P(0)=10^{C}=80[/tex]
Applying logarithmic properties:
[tex]P=10^{0.1t+C}=10^{0.1t}*10^{C}[/tex]
So, the final function is:
[tex]P(t)=80(10^{0.1t})[/tex]
The population size over time follows an exponential growth function described by P(t) = 80 * [tex]e^(0.23026t)[/tex], where the slope on a semilog plot of 0.1 is converted to the natural log base to find the growth rate constant.
When dealing with population growth, plotting the data on a semilog plot where the logarithm of the population size is plotted against time can simplify the analysis. If the resulting plot is a straight line with a slope of 0.1, this indicates exponential growth, because plotting an exponential function on a semilog plot yields a straight line.
The general form of an exponential growth function is:
P(t) = P0 * [tex]e^{kt}[/tex]Where:
P(t) is the population at time tP0 is the initial population sizek is the growth rate constantt is timeGiven that the slope of the line on the semi log plot is 0.1, this slope represents the constant k in the context of logs with base 10.
To convert this to a natural logarithm base (e), use the conversion factor ln(10):
k = 0.1 * ln(10)
Since ln(10) ≈ 2.3026, we have:
k = 0.1 * 2.3026
≈ 0.23026
Given the initial population size (P0) at time t = 0 is 80, the function describing the population size over time t is:
P(t) = 80 * [tex]e^(0.23026t)[/tex],
Using the Chinese Remainder Theorem, solve the congruence
x 15 (mod 42)
x 5 (mod 19)
19 and 42 are coprime, so we can use the CRT right away. Start with
[tex]x=19+42[/tex]
Taken mod 42, we're left with a remainder of 19. Notice that
[tex]19\cdot3\equiv57\equiv15\pmod{42}[/tex]
so we need to multiply the first term by 3 to get the remainder we want.
[tex]x=19\cdot3+42[/tex]
Next, taken mod 19, we're left with a remainder of 4. Notice that
[tex]42\cdot6\equiv252\equiv5\pmod{19}[/tex]
so we need to multiply the second term by 6.
Then by the CRT, we have
[tex]x\equiv19\cdot3+42\cdot6\equiv309\pmod{42\cdot19}\implies x\equiv309\pmod{798}[/tex]
so that any solution of the form [tex]x=798n+309[/tex] is a solution to this system.
A group of research proposals was evaluated by a panel of experts to decide whether or not they were worthy of funding. When these same proposals were submitted to a second independent panel of experts, the decision to fund was reversed in 30% of the cases. If the probability that a proposal is judged worthy of funding by the first panel is .2, what are the probabilities that: a.A worthy proposal is approved by both panels. b.A worthy proposal is disapproved by both panels. c.A worthy proposal is approved by one panel.
Explanation of probabilities for a worthy research proposal approved by two panels, disapproved by two panels, and approved by one panel.
A worthy proposal is approved by both panels:
Probability = (Probability approved by Panel 1) × (Probability approved by Panel 2) = 0.2 × 0.7 = 0.14
A worthy proposal is disapproved by both panels:
Probability = (Probability disapproved by Panel 1) × (Probability disapproved by Panel 2) = 0.8 × 0.3 = 0.24
A worthy proposal is approved by one panel:
Probability = (Probability approved by Panel 1 and disapproved by Panel 2) + (Probability disapproved by Panel 1 and approved by Panel 2) = (0.2 × 0.3) + (0.8 × 0.7) = 0.06 + 0.56 = 0.62
Assume that women's heights are normally distributed with a mean given by mu equals 62.5 in, and a standard deviation given by sigma equals 2.5 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 63 in. (b) If 35 women are randomly selected, find the probability that they have a mean height less than 63 in. (a) The probability is approximately nothing. (Round to four decimal places as needed.) (b) The probability is approximately nothing. (Round to four decimal places as needed.)
Answer: a) The probability is approximately = 0.5793
b) The probability is approximately=0.8810
Step-by-step explanation:
Given : Mean : [tex]\mu= 62.5\text{ in}[/tex]
Standard deviation : [tex]\sigma = \text{2.5 in}[/tex]
a) The formula for z -score :
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Sample size = 1
For x= 63 in. ,
[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{1}}}=0.2[/tex]
The p-value = [tex]P(z<0.2)=[/tex]
[tex]0.5792597\approx0.5793[/tex]
Thus, the probability is approximately = 0.5793
b) Sample size = 35
For x= 63 ,
[tex]z=\dfrac{63-62.5}{\dfrac{2.5}{\sqrt{35}}}\approx1.18[/tex]
The p-value = [tex]P(z<1.18)[/tex]
[tex]= 0.8809999\approx0.8810[/tex]
Thus , the probability is approximately=0.8810.
Compute the surface integral over the given oriented surface: F=y3i+8j−xk, portion of the plane x+y+z=1 in the octant x,y,z≥0, downward-pointing normal
We are to find the surface integral of [tex]\mathbf{\int \int _S \ F.ds}[/tex]
where;
the surface of the portion of the plane is [tex]x+y+z =1[/tex]; in the 1st octant, [tex]x,y,z \geq 0[/tex] , and:the oriented surface [tex]\mathbf{F = y^3i+8j-xk}[/tex]∴
The plane x + y + z = 1
z = 1 - x - y
[tex]\dfrac{\partial z}{\partial x} = -1[/tex]
[tex]\dfrac{\partial z}{\partial y} = -1[/tex]
Since the surface is oriented downward
∴
[tex]dS = \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy[/tex]
[tex]dS = (-i-j-k) dxdy[/tex]
However,
Flux = [tex]\mathbf{\int \int _S \ F.ds}[/tex]
[tex]= \int \int_R F. \Big( \dfrac{\partial z}{\partial x}i + \dfrac{\partial z}{\partial y}j - k) dxdy \\ \\ \\ = \int^1_{ x=0} \int ^{1-x}_{y=0} ( y^3i + 8j -xk)*(-i-j-k) dx dy \\ \\ \\ =\int^1_{ x=0} \int ^{1-x}_{y=0} (-y^3 -8+x) dxdy \\ \\ \\ = \int^1_{ x=0} \Big( \int ^{1-x}_{y=0} \Big(x-y^3 -8\Big) dy \Big) dx \\ \\ \\ = \int^1_{ x=0} \Bigg [xy - \dfrac{y^4}{4}-8y \Bigg]^{1-x}_{y=0} \ dx \\ \\ \\[/tex]
[tex]= \int^1_{ x=0} \Bigg [x(1-x) - \dfrac{1}{4}(1-x)^{4} - 8(1-x) \Bigg ] dx \\ \\ \\ = \int^1_{ x=0} \Bigg [(x-x^2) - \dfrac{1}{4}(x-1)^4+8(x-1)\Bigg] dx[/tex]
[tex]= \Bigg [ \dfrac{x^2}{2}-\dfrac{x^3}{3} - \dfrac{1}{4} \dfrac{(x-1)^5}{5}+(x-1)^2\Bigg] ^1_0[/tex]
[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} -0+0-0+0+ \dfrac{1}{4}\times \dfrac{(-1)^5}{5}-(0-1)^2\Bigg][/tex]
[tex]= \Bigg [ \dfrac{1}{2}-\dfrac{1}{3} - \dfrac{1}{20}-1\Bigg][/tex]
[tex]= \Bigg [ \dfrac{30-20-3-60}{60}\Bigg][/tex]
[tex]= \Bigg [ \dfrac{-53}{60}\Bigg][/tex]
Therefore, we can conclude that the surface integral is [tex]\mathbf{-\dfrac{53}{60}}[/tex]
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To compute a surface integral, take the antiderivatives of both dimensions defining the area with the surface edges as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.
Explanation:A surface integral over the given oriented surface can be computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface as the bounds of the integral. The net flux through the surface can be found using the open surface integral formula.
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Given A = {3,4, 13, 15) and B = { 13, 15, 21, 25, 30, 37). Which of the following are true? I. AUB 13,4,21,25, 30, 37) III. ACB II. AnB-(13, 15) a) I and III only b) OII only e) Oll and III only d) I and II only e) OI, II, and III None of the above
The correct statements regarding the sets A and B is:
II.
[tex]A\bigcap B=\{13,15\}[/tex]
Step-by-step explanation:We are given set A as:
[tex]A=\{3,4,13,15\}[/tex]
and set B as:
[tex]B=\{13,15,21,25,30,37\}[/tex]
and the three statements are given by:
I.
[tex]A\bigcup B=\{12,4,21,25,30,37\}[/tex]
II.
[tex]A\bigcap B=\{13,15\}[/tex]
III.
[tex]A\subset B[/tex]
We know that the union of two sets is the collection of all the elements which belong either to A or to B.i.e. from the given sets we have:
[tex]A\bigcup B=\{3,4,13,15,21,25,30,37\}[/tex]
Also, the intersection of two sets is the collection of all the elements which belongs to both the set.i.e.
[tex]A\bigcap B=\{13,15\}[/tex]
Also, we know that A is a subset of B if:All the elements of A are contained in B.
But from the given sets we see that:
3,4 belongs to A but they do not belong to B.
Hence, A is not a subset of B.
A candy company that makes colored candies claims that 10% of the candies it produces are green and that bags are packed randomly. In a really large bag of candies, we found 12% of 500 candies were green. Is this evidence that the manufacturing process is out of control and has made too many greens? Explain.
No. The sample size is too small to be reliable. If there were many bags of 500 candies that consistently showed a larger-than-marketed amount of green candy, that will be better proof.
The difference between the observed proportion of green candies (0.12) and candy company's claimed proportion (0.10) does not necessarily mean that the manufacturing process is out of control. It could simply be down to natural variation or chance. A definitive conclusion would require a proper statistical test using more data.
Explanation:In this scenario, we are looking at a statistic known as the proportion. It is calculated by dividing the number of successful outcomes (green candies) by the total number of outcomes (total candies). The candy company claims a proportion of 10%, which is 0.10. However, the observed proportion in your large bag of candies was 12% of 500, which works out to 0.12.
From a statistical standpoint, a slight variation from the claimed percentage is not unusual, as there will always be some random variability due to chance. While the difference between the observed proportion (0.12) and the expected proportion (0.10) might seem significant at first glance, we cannot definitively say from this one sample that the manufacturing process is out of control without conducting a proper hypothesis test — ideally, using additional data from more bags of candy.
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You have 100 quarters in a jar. One of the quarters is double sided (heads). You pick out a random quarter and flip it 7 times, and get all heads. what is the probability you picked the double sided quarter? Then, given that you flipped it 7 times with all heads, what is the probability that you'll get heads on the 8th flip?
Final answer:
The probability of picking the double-sided quarter is 1/100. The probability of getting heads on the 8th flip given that the previous 7 flips were all heads is still 1/2.
Explanation:
Let's calculate the probability of picking the double-sided (heads) quarter first.
Out of the 100 quarters, only 1 is double-sided. Therefore, the probability of picking the double-sided quarter is 1/100.
Now, let's calculate the probability of getting heads on the 8th flip given that the previous 7 flips were all heads.
The probability of getting heads on a single flip is 1/2, since there are two possible outcomes (heads or tails) and both are equally likely.
Since the flips are independent, the probability of getting heads on the 8th flip given that the previous 7 flips were all heads is still 1/2.
Suppose that a department contains 9 men and 15 women. How many different committees of 6 members are possible if the committee must have strictly more women than men?
Answer: The required number of different possible committees is 81172.
Step-by-step explanation: Given that a department contains 9 men and 15 women.
We are to find the number of different committees of 6 members that are possible if the committee must have strictly more women than men.
Since we need committees of 6 members, so the possible combinations are
(4 women, 2 men), (5 women, 1 men) and (6 women).
Therefore, the number of different committees of 6 members is given by
[tex]n\\\\\\=^{15}C_4\times ^9C_2+^{15}C_5\times ^9C_1+^{15}C_6\\\\\\=\dfrac{15!}{4!(15-4)!}\times \dfrac{9!}{2!(9-2)!}+\dfrac{15!}{5!(15-5)!}\times \dfrac{9!}{1!(9-1)!}+\dfrac{15!}{6!(15-6)!}\\\\\\\\=\dfrac{15\times14\times13\times12\times11!}{4\times3\times2\times1\times11!}\times\dfrac{9\times8\times7!}{2\times1\times7!}+\dfrac{15\times14\times13\times12\times11\times10!}{5\times4\times3\times2\times1\times10!}\times\dfrac{9\times8!}{1\times8!}+\dfrac{15\times14\times13\times12\times11\times10\times9!}{6\times5\times4\times3\times2\times1\times9!}\\\\\\=1365\times36+3003\times9+5005\\\\=49140+27027+5005\\\\=81172.[/tex]
Thus, the required number of different possible committees of 6 members is 81172.
A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function c(x) = 0.3x^2 - 156x + 26,657 . How many machines must be made to minimize the unit cost?
Do not round your answer.
Answer:
260 machines for minimum cost.
Step-by-step explanation:
c(x) = 0.3x^2 - 156x + 26.657
Finding the derivative:
c'(x) = 0.6x - 156
0.6x - 156 = 0 for maxm/minm cost.
x = 156 / 0.6
= 260
The second derivative is positive (0.6) so this is a minimum.
Answer:
[tex]x=260\ machines[/tex]
Step-by-step explanation:
Note that we have a cudratic function of negative principal coefficient.
The minimum value reached by this function is found in its vertex.
For a quadratic function of the form
[tex]ax ^ 2 + bx + c[/tex]
the x coordinate of the vertex is given by the following expression
[tex]x=-\frac{b}{2a}[/tex]
In this case the function is:
[tex]C(x) = 0.3x^2 - 156x + 26,657[/tex]
So:
[tex]a=0.3\\b=-156\\c=26,657[/tex]
Then the x coordinate of the vertex is:
[tex]x=-\frac{-156}{2(0.3)}[/tex]
[tex]x=260\ machines[/tex]
Then the number of machines that must be made to minimize the cost is:
[tex]x=260\ machines[/tex]
The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $10. If 300 utility bills are randomly selected from this city, approximately how many of them will be more than $115?
First compute the probability that a bill would exceed $115. Let [tex]X[/tex] be the random variable representing the value of a monthly utility bill. Then transforming to the standard normal distribution we have
[tex]Z=\dfrac{X-100}{10}[/tex]
[tex]P(X>115)=P\left(Z>\dfrac{115-100}{10}\right)=P(Z>1.5)\approx0.0668[/tex]
Then out of 300 randomly selected bills, one can expect about 6.68% of them to cost more than $115, or about 20.
Use the method of reduction of order to find a second solution to t^2y' + 3ty' – 3y = 0, t> 0 Given yı(t) = t y2(t) = Preview Give your answer in simplest form (ie no coefficients)
Let [tex]y_2(t)=tv(t)[/tex]. Then
[tex]{y_2}'=tv'+v[/tex]
[tex]{y_2}''=tv''+2v'[/tex]
and substituting these into the ODE gives
[tex]t^2(tv''+2v')+3t(tv'+v)-3tv=0[/tex]
[tex]t^3v''+5t^2v'=0[/tex]
[tex]tv''+5v'=0[/tex]
Let [tex]u(t)=v'(t)[/tex], so that [tex]u'(t)=v''(t)[/tex]. Then the ODE is linear in [tex]u[/tex], with
[tex]tu'+5u=0[/tex]
Multiply both sides by [tex]t^4[/tex], so that the left side can be condensed as the derivative of a product:
[tex]t^5u'+5t^4u=(t^5u)'=0[/tex]
Integrating both sides and solving for [tex]u(t)[/tex] gives
[tex]t^5u=C\implies u=Ct^{-5}[/tex]
Integrate again to solve for [tex]v(t)[/tex]:
[tex]v=C_1t^{-6}+C_2[/tex]
and finally, solve for [tex]y_2(t)[/tex] by multiplying both sides by [tex]t[/tex]:
[tex]tv=y_2=C_1t^{-5}+C_2t[/tex]
[tex]y_1(t)=t[/tex] already accounts for the [tex]t[/tex] term in this solution, so the other independent solution is [tex]y_2(t)=t^{-5}[/tex].
A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 46.0 and 56.0 minutes. Find the probability that a given class period runs between 50.75 and 51.25 minutes.
Answer: 0.05
Step-by-step explanation:
Given : Interval for uniform distribution : [46.0 minutes, 56.0 minutes]
The probability density function will be :-
[tex]f(x)=\dfrac{1}{56-46}=\dfrac{1}{10}=0.1\ \ , 46<x<56[/tex]
The probability that a given class period runs between 50.75 and 51.25 minutes is given by :-
[tex]P(50.75<x<51.25)=\int^{51.25}_{50.75}f(x)\ dx\\\\=(0.1)[x]^{51.25}_{50.75}\\\\=(0.1)(51.25-50.75)=0.05[/tex]
Hence, the probability that a given class period runs between 50.75 and 51.25 minutes = 0.05
A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution how many minutes will it take for the medicine to be given
Answer:
433 minutes
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
The nurse is infusing 1.5 ml of solution every x minutes, Since there are a total of 650 ml we can divide the amount the nurse is infusing every x minutes by the total in order to calculate the amount of time it will take the nurse to finish.
[tex]\frac{650}{1.5} = 433.33min[/tex]
So we can see that it will take the nurse 433 min to infuse all of the solution.
Answer:
In 433.33 minutes will it take for the medicine to be given.
Step-by-step explanation:
Given : A nurse must infuse 1.5 ml of solution in x minutes and she has 650 ml of solution.
To find : How many minutes will it take for the medicine to be given ?
Solution :
As 1.5 ml of solution infuse in x minutes
i.e. the amount of solution she has = [tex]1.5x[/tex]
Now, she has 650 ml of solution.
which means [tex]1.5x=650[/tex]
Solving the equation,
[tex]x=\frac{650}{1.5}[/tex]
[tex]x=433.33[/tex]
Therefore, in 433.33 minutes will it take for the medicine to be given.
Two teams are engaged in a tug-of-war. According to Newton’s third law, the red team pulls on the blue team with the same magnitude of force that the blue team pulls on the red team. How, then, can one team win?
Both the teams are held at their place by the friction between the ground and their feet. If the pulling force as mentioned above exceeds this friction, the losing team experiences a net force and accelerates.
What is Newton’s third law?If object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on the object.
When a team pulls the rope, the rope pulls the team.
This was Newton's third law.
When both teams pull in opposite directions, they often don't exert equal forces; the stronger team exerts a larger force.
So the two forces don't balance each other and the rope accelerates towards the stronger team.
The weaker team experiences a net force pulling them.
This force is the sum of (i)The reaction by the rope as mentioned above and (ii) The friction between the rope and the team members' hands.
Both the teams are held at their place by the friction between the ground and their feet. If the pulling force as mentioned above exceeds this friction, the losing team experiences a net force and accelerates.
To know more about Newton's third law click the link given below.
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Newton's Third Law explains the forces exerted in a tug-of-war, but the winning team is often determined by factors like friction and the angle of pulling. Greater friction helps a team resist the opposing force, while a lower angle of pulling may provide an advantage in distributing the tug's force.
Explanation:Even though Newton's Third Law of Motion states that for every action there is an equal and opposite reaction, this does not mean that all forces will cancel each other out. In the context of a tug-of-war, friction and the angle in which the tug is exerted can influence outcomes.
The tug-of-war is not only a battle of forces but also a battle against friction. Each team is pulling as hard as they can in opposite directions, establishing the action-reaction pair. However, their ability to maintain their ground or pull the other team relies significantly on the friction between their feet and the ground. If either team can increase this friction (e.g., by wearing shoes with better grip, or planting their feet more effectively), they can resist the pulling force of the other team, leading to their victory.
Moreover, the angle at which a team pulls the rope can also determine the advantage in a tug of war. Teams that manage to keep the rope lower to the ground distribute the force more efficiently and therefore have a greater chance of winning.
Learn more about Tug-of-War Forces here:https://brainly.com/question/33722744
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Additional Proofs: Prove each statement below using Proof by Contradiction. 1. The sum of any rational number and any irrational number is irrational. S 2. For all integers m, if m is even, then 3m+7 is odd I integer, then, 2a +3b is even 3. If a is any odd integer, and b is any even 4. Let A and B be sets from a universe U. If Bc A, then AC B
Answer:
See below.
Step-by-step explanation:
1. Suppose that the sum is rational then we can write:
a/b + i = c/d where i is irrational and by definition a/b and c/d are rational.
Rearranging:
i = c/d - a/b
Now the sum on the right is rational so 'irrational' = 'rational' which is a contradiction.
So the original supposition is false and the sum must be irrational.
2. Proof of For all integers m if m is even then 3m + 7 is odd:
If m is even then 3m is even.
Suppose 3m + 7 is even, then:
3m + 7 = 2p where p is an integer.
3m - 2p = -7
But 3m and 2p are both even so their result is even and -7 is odd.
Therefore the original supposition is false because it leads to a contradiction, so 3m + 7 is odd.
1 Kg = 2.2046 pounds what will be the answer in round to nearest tenth 4550 pounds = ______ kg
Answer:
2063.9 kg
Step-by-step explanation:
Given,
1 kg = 2.2046 pounds
This can also be written as:
1 pound = 1/2.2046 kg
We have to calculate the value of 4550 pounds in kg up to nearest 10th
Thus,
[tex]4550 pounds= \frac{4550}{2.2046} kg[/tex]
Solving the above equation, we get:
4550 pounds = 2063.8664 kg
Rounding the above result to nearest tenth as:
4550 pounds = 2063.9 kg
Determine whether or not the vector field is conservative. If it is, find a function f such that F = ∇f. If the vector field is not conservative, enter NONE. F(x, y, z) = 5y2z3 i + 10xyz3 j + 15xy2z2 k f(x, y, z) = + K
[tex]\vec F[/tex] is conservative if we can find a scalar function [tex]f[/tex] such that [tex]\nabla f=\vec F[/tex]. This would require
[tex]\dfrac{\partial f}{\partial x}=5y^2z^3[/tex]
[tex]\dfrac{\partial f}{\partial y}=10xyz^3[/tex]
[tex]\dfrac{\partial f}{\partial z}=15xy^2z^2[/tex]
Integrate both sides of the first PDE with respect to [tex]x[/tex]:
[tex]f(x,y,z)=5xy^2z^3+g(y,z)[/tex] (*)
Differentiate both sides of (*) with respect to [tex]y[/tex]:
[tex]\dfrac{\partial f}{\partial y}=10xyz^3=10xyz^3+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)[/tex]
Differentiate both sides of (*) with respect to [tex]z[/tex]:
[tex]\dfrac{\partial f}{\partial z}=15xy^2z^2=15xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]
So we have
[tex]f(x,y,z)=5xy^2z^3+C[/tex]
and so [tex]\vec F[/tex] is indeed conservative.
Sue a tollbooth worker is paid $12.50 per hour between 8 Am and 4 pm. After 4 pm she is paid $14.25 per hour. If Sue works from 10 am to 6 pm one particular day how much money does she earn that day?
Answer:
She earns that day $103.5
Step-by-step explanation:
Lets explain how to solve the question
- She is paid $12.50 per hour between 8 am and 4 pm
- After 4 pm she is paid $14.25 per hour
- She works one particular day from 10 am to 6 pm
* Lets distribute her time of work into two part according to her
payment above
# 1st part from 10 am to 4 pm
∵ She is paid $12.50 per hour between 8 am to 4 pm
∵ She works from 10 am to 4 pm
- Calculate how many hours between 10 am and 4 pm
∵ am and pm not like terms in time, so change the time to 24-hours
∵ 4 pm in 24-hour time is 4 + 12 = 16
∴ She works form 10 to 16
∴ She works for 6 hours (16 - 10 = 6)
∴ She is paid = 12.50 × 6 = $75
# 2nd part after 4 pm to 6 pm
∵ She is paid $14.25 per hour after 4 pm
∵ She works from 4 pm to 6 pm
- Calculate how many hours between 4 pm and 6 pm
∵ Both times are pm
∴ She works for 2 hours (6 - 4 = 2)
∴ She is paid = 14.25 × 2 = $28.5
- Add the two answers of the 1st part and the 2nd part
∴ She earns = 75 + 28.5 = $103.5
* She earns that day $103.5
Answer:
$103.50.
Step-by-step explanation:
From 10 am until 4 pm ( 6 hours) she earns 6 * 12.50 = $75.00.
Then from 4 pm to 6 pm she earns 2*14.25 = $28.50.
Total for that day = 75 + 28.50
= $103.50.
Solve the congruence 2x 3 (mod 11):
[tex]2x\equiv3\pmod{11}[/tex]
Since 3 * 4 = 12 = 1 mod 11, we can multiply both sides by 4 to get
[tex]4\cdot2x\equiv3\cdot4\pmod{11}\implies8x\equiv1\pmod{11}[/tex]
Since 8 * 7 = 56 = 1 mod 11, multiplying both sides by 7 gives
[tex]7\cdot8x\equiv7\cdot1\pmod{11}\implies x\equiv7\pmod{11}[/tex]
Which expressions can be used to find m angle ABC? Select
two options.
Answer:
cos ⁻¹ [6.3/9.8] . . . second choice, andsin ⁻¹ [7.5/9.8] . . . last choice.Explanation:
The little red square indicates that the angle C measures 90°, which means that ABC is a right triangle.
The opposite side to angle C is the hypotenuse of the triangle. Then, the hypotenuse measures 9.8 in.
The requested measure, m ∠ ABC, is the measure of the angle B.
With respect to angle B you have:
Opposite leg: 7.5 inAdjacent leg: 6.3 inThus, the trigonometric ratios for angle B are:
cos B = adjacent leg / hypotenuse = 6.3 in / 9.8 in = 6.3 / 9.8sin B = opposite leg / hypotenuse = 7.5 in / 9.8 in = 7.5 / 9.8Finally, to find the measures of angle B, you just need to find the inverse functions, which are:
cos ⁻¹ [6.3/9.8] . . . second choice, andsin ⁻¹ [7.5/9.8] . . . last choice.Answer:A&E
Step-by-step explanation:
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