Answer:
1.2 grams.
Step-by-step explanation:
We have been given that an oral concentrate of sertraline hydro-chloride (Zoloft) contains 20 mg/mL of the drug.
First of all, we will find number of mg in 60 mL container of the concentrate as:
[tex]\frac{\text{20 mg}}{\text{ml}}\times \text{60 ml}[/tex]
[tex]\text{20 mg}\times 60[/tex]
[tex]\text{1200 mg}[/tex]
We know 1 gram equals 1000 mg.
[tex]\text{1200 mg}\times \frac{\text{1 gram}}{\text{1000 mg}}[/tex]
[tex]1.2\times\text{1 gram}[/tex]
[tex]1.2\text{ grams}[/tex]
Therefore, 1.2 grams of sertraline hydrochloride are in each 60-mL container of the concentrate.
elena has 60 colored pencils, and lucy has 26 colored pencils. howmany pencils must elena give to lucy so that elena will have 4 more colored pencils than lucy?
Answer: 15
Step-by-step explanation:
Given : Elena has 60 colored pencils, and Lucy has 26 colored pencils.
Total colored pencils = [tex]60+26=86[/tex]
Let 'x' denotes the number of pencils Lucy has when she get pencils from Elena , and Elena left with 'y' pencils.
Then, by considering the given information we have the following system :-
[tex]\text{Total pencils : }x+y=86-----(1)\\\\\text{Colored pencils left with Elena : }y=4+x-----(2)[/tex]
We can rewritten the equation (2) as
[tex]y-x=4[/tex]
Now, add equation (1) and (2), we get
[tex]2y=90\\\\\Rightarrow\ y=\dfrac{90}{2}=45[/tex]
Put the value of y in (2), we get
[tex]45=4+x\\\\\Rightarrow\ x=45-4=41[/tex]
It means, the number of pencils Lucy has now =41
The number of pencils Elena give to Lucy= [tex]41-26=15[/tex]
Hence, Elena must give 15 pencils to lucy so that elena will have 4 more colored pencils than Lucy.
A recent study found a correlation between gum disease and heart disease. This result indicates that gum disease causes people to develop heart disease.
True or False?
Answer:
False
Step-by-step explanation:
Correlation measures the strength of the relation between two variables.
Further, Correlation is said to be positive if increasing/decreasing the one variable, also increases/decreases the values of another variable.
Correlation is said to be negative if increasing/decreasing the one variable, also decreases/increases the values of another variable.
Since we don't know here exists a positive correlation or negative correlation.
So here are two possible conditions:
The person who has Gum disease also has heart disease.
And, the person has Gum disease can never have heart disease.
Thus, the given statement is false.
The statement is false because a correlation found in a study does not necessarily mean one factor (gum disease) is the cause of the other (heart disease). The cause and effect relationship must be established through further studies.
Explanation:The statement 'A recent study found a correlation between gum disease and heart disease. This result indicates that gum disease causes people to develop heart disease.' is False. A correlation implies a relationship between two elements, but it does not indicate a cause and effect relationship.
This means although the study shows a link or association between gum disease and heart disease, it does not mean gum disease causes heart disease. It could be that people with poor gum health also tend to have poor overall health including heart health. Alternatively, there could be a third underlying factor that leads to both conditions. Therefore, the cause and effect relationship must be established through further studies.
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Margo borrows $900, agreeing to pay it back with 7% annual interest after 9 months. How much interest will she pay? Round your answer to the nearest cent, if necessary.
Answer:
[tex]I=\$47.25[/tex]
Step-by-step explanation:
we know that
The simple interest formula is equal to
[tex]I=P(rt)[/tex]
where
I is the Final Interest Value
P is the amount of money borrowed
r is the rate of interest
t is Number of Time Periods
in this problem we have
[tex]t=9/12\ years\\ P=\$900\\r=0.07[/tex]
substitute in the formula above
[tex]I=900(0.07(9/12))[/tex]
[tex]I=\$47.25[/tex]
Final answer:
Margo will pay $47.25 in interest on a $900 loan with a 7% annual interest rate after 9 months, after converting the rate to a decimal and the time to years for the calculation.
Explanation:
The question asks how much interest Margo will pay on a $900 loan with an annual interest rate of 7% after 9 months. To calculate this, we need to know that Interest = Principal × Rate × Time, where the principal is the amount borrowed, the rate is the annual interest rate (as a decimal), and the time is the period of the loan in years
First, convert the annual interest rate from a percentage to a decimal by dividing by 100: 7% / 100 = 0.07. Next, convert the loan period from months to years since the interest rate is annual. There are 12 months in a year, so 9 months is equal to 9/12 or 0.75 years.
Now, calculate the interest: $900 (Principal) × 0.07 (Rate) × 0.75 (Time) = $47.25. Therefore, Margo will pay $47.25 in interest.
Find the equation of the line that pass through the points (4,8) and (6,2
Answer:
3x + y = 20
Step-by-step explanation:
The equation of line passing through two points is determined by formula:
[tex]y-y_{1}=\frac{y_{2}-y_{1}}{x_{2} - x_{1}}(x - x_{1})[/tex]
Here, (x₁ , y₁) = (4, 8)
and (x₂, y₂) = (6, 2)
Putting these value in above formula. We get,
[tex]y-8=\frac{2-8}{6 - 4}(x - 4)[/tex]
⇒ [tex]y-8=\frac{-6}{2}(x - 4)[/tex]
⇒ ( y - 8) = -3 (x - 4)
⇒ y - 8 = -3x + 12
⇒ 3x + y = 20
which is required equation.
A three inch diameter pulley on an electric motor that runs at 1800
revolutions per minute is connected by a belt to a six inch
diameter pullley on a saw arbor.
angular speed = central angle/time, arc length = (central
angle)(radius)
a. Find the angular speed (in radians per minute) of each. ( 3 in
and 6 in pully)
b. find the revolutions per minute of the saw.
Answer:
a) 3 inch pulley: 11,309.7 radians/min
6) 6 inch pulley: 5654.7 radians/min
b) 900 RPM (revolutions per minute)
Step-by-step explanation:
Hi!
When a pulley wirh radius R rotantes an angle θ, the arc length travelled by a point on its rim is Rθ. Then the tangential speed V is related to angular speed ω as:
[tex]V=R\omega[/tex]
When you connect two pulleys with a belt, if the belt doesn't slip, each point of the belt has the same speed as each point in the rim of both pulleys: Then, both pulleys have the same tangential speed:
[tex]\omega_1 R_1 = \omega_2 R_2\\[/tex]
[tex]\omega_2 = \omega_1 \frac{R_1}{R_2} =1800RPM* \frac{3}{6}= 900RPM[/tex]
We need to convert RPM to radias per minute. One revolution is 2π radians, then:
[tex]\omega_1 = 1800*2\pi \frac{radians}{min} = 11,309.7\frac{radians}{min}[/tex]
[tex]\omega_2 = 5654.7 \frac{radians}{min}[/tex]
The saw rotates with the same angular speed as the 6 inch pulley: 900RPM
a. The angular speed of the 3 inch pulley is 3600π radians/min and the angular speed of the 6 inch pulley is 7200π radians/min. b. The revolutions per minute of the saw is 900.
Explanation:a. To find the angular speed in radians per minute, we need to convert the revolutions per minute to radians per minute. Since 1 revolution is equal to 2π radians, we can calculate the angular speed of the 3 inch pulley as follows:
Angular speed = (Revolutions per minute) x (2π radians per revolution)
Angular speed = (1800 rev/min) x (2π radians/rev) = 3600π radians/min
Similarly, for the 6 inch pulley:
Angular speed = (Revolutions per minute) x (2π radians per revolution)
Angular speed = (1800 rev/min) x (2π radians/rev) = 7200π radians/min
b. To find the revolutions per minute of the saw, we need to use the ratio of the diameters of the two pulleys. Since the diameter of the 6 inch pulley is twice the diameter of the 3 inch pulley, the revolutions per minute of the saw will be half of the revolutions per minute of the motor. Therefore, the revolutions per minute of the saw is 900.
3/4x -1/2y= 8 and 2x +y=40
Answer:
The value of x is 16 and the value of y is 8.
Step-by-step explanation:
Consider the provided equation.
[tex]\frac{3}{4}x -\frac{1}{2}y= 8\ and\ 2x +y=40[/tex]
Isolate x for [tex]\:\frac{3}{4}x-\frac{1}{2}y=8[/tex]
[tex]\frac{3}{4}x-\frac{1}{2}y+\frac{1}{2}y=8+\frac{1}{2}y[/tex]
[tex]\frac{3}{4}x=8+\frac{1}{2}y[/tex]
Multiply both side by 4 and simplify.
[tex]3x=32+2y[/tex]
[tex]x=\frac{32+2y}{3}[/tex]
Substitute the value of x in [tex]2x +y=40[/tex]
[tex]2\cdot \frac{32+2y}{3}+y=40[/tex]
[tex]\frac{64}{3}+\frac{7y}{3}=40[/tex]
[tex]64+7y=120[/tex]
[tex]7y=56[/tex]
[tex]y=8[/tex]
Now substitute the value of y in [tex]x=\frac{32+2y}{3}[/tex]
[tex]x=\frac{32+2\cdot \:8}{3}[/tex]
[tex]x=16[/tex]
Hence, the value of x is 16 and the value of y is 8.
A business magazine conducted a survey of 751 employees who had been at their current employer for 5 or more years. Of these employees, 295 responded that they were bored in their current position. Create a 99% confidence interval for the proportion of employees who have been with their current employer for 5 or more years and reported being bored in their current position. Use Excel to create the confidence interval, rounding to four decimal places.
Answer:
the [tex]95\%[/tex] confidence interval for the population proportion is:
[tex]\left [0.3469, \hspace{0.1cm} 0.4387\right][/tex]
Step-by-step explanation:
To solve this problem, a confidence interval of [tex](1-\alpha) \times 100\%[/tex] for the population proportion will be calculated.
[tex]$$Sample proportion: $\bar P=0.3928$\\Sample size $n=751$\\Confidence level $(1-\alpha)\times100\%=99\%$\\$\alpha: \alpha=0.01$\\Z values (for a 99\% confidence) $Z_{\alpha/2}=Z_{0.005}=2.5758$\\\\Then, the (1-\alpha) \times 100\%$ confidence interval for the population proportion is given by:\\\\\left [\bar P - Z_{\frac{\alpha}{2}}\sqrt{\frac{\bar P(1- \bar P)}{n}}, \hspace{0.3cm}\bar P + Z_{\frac{\alpha}{2}}\sqrt\frac{\bar P(1- \bar P)}{n} \right ][/tex]
Thus, the [tex]95\%[/tex] confidence interval for the population proportion is:
[tex]\left [0.3928 - 2.5758\sqrt{\frac{0.3928(1-0.3928)}{751}}, \hspace{0.1cm}0.3928 + 2.5758\sqrt{\frac{0.3928(1-0.3928)}{751}} \right ]=\left [0.3469, \hspace{0.1cm} 0.4387\right][/tex]
Which of the following sets are equal to {x | x < 9 and x >2}
Question 5 options:
{3, 4, 5, 6, 7, 8}
{2, 3, 4, 5, 6, 7, 8, 9}
{8, 7, 6, 5, 3}
{ }
{2, 3, 4, 5, 6, 7}
Answer:
{3, 4, 5, 6, 7, 8}
Step-by-step explanation:
Integers that are less than 9 and greater than 2 include the integers 3 through 8.
The correct set equal to {x | x < 9 and x > 2} is {3, 4, 5, 6, 7, 8}, as it includes all the integers that satisfy the given condition.
The given set is {x | x < 9 and x > 2}, which translates to all numbers greater than 2 and less than 9. When comparing this to the options provided, we need to ensure that the numbers within the set are all and only the integers that satisfy these conditions, regardless of their order. The set {3, 4, 5, 6, 7, 8} matches this description exactly, as it includes all the integers that are greater than 2 and less than 9. Sets in mathematics do not consider the order of elements; they only consider the presence of elements. Therefore, the correct option that is equal to the given set is {3, 4, 5, 6, 7, 8}.
Compute the exact interest on $5,870 at 12% if the money is borrowed from June to December of the same year.
Answer:
The exact interest on $5,870 at 12% is $58.70.
Step-by-step explanation:
Given information:
Principal = $5870
Interest rate = 12% = 0.12
Time = June-December = 7 months.
We know that
1 year = 12 months
1/12 year = 1 month
7/12 year = 7 month
Time = 7/12 year
Formula for simple interest:
[tex]I=P\times r\times t[/tex]
where, P is principal, r is rate of interest and t is time in years.
Substitute P=5870, r=0.12 and t=7/12 in the above formula.
[tex]I=5870\times 0.12\times \frac{1}{12}[/tex]
[tex]I=5870\times 0.01[/tex]
[tex]I=58.70[/tex]
Therefore the exact interest on $5,870 at 12% is $58.70.
An FM radio station broadcasts at 98 MHz. what is the energy of each photon in Joule? Use h= 6.6 X10^-34 J*s for Planck constant.
Answer:
The energy of each photon is [tex]6.468 \times 10^{-26}[/tex] Joule.
Step-by-step explanation:
Consider the provided information.
According to the plank equation:
[tex]E=h\nu[/tex]
Where E is the energy of photon, h is the plank constant and [tex]\nu[/tex] is the frequency.
It is given that [tex]h= 6.6 \times10^{-34}[/tex] and [tex]\nu=98MHz[/tex]
98Mhz = [tex]98\times 10^6Hz[/tex]
Substitute the respective value in plank equation.
[tex]E=6.6\times 10^{-34}\times 98\times 10^6[/tex]
[tex]E=6.6\times 98\times 10^{-34+6}[/tex]
[tex]E=646.8 \times10^{-28}[/tex]
[tex]E=6.468 \times 10^{-26}[/tex]
Hence, the energy of each photon is [tex]6.468 \times 10^{-26}[/tex] Joule.
Solve the following problems manually or using the MS Excel
Mohamed has enough wood to make 24 small chairs or 6 large tables. In other words, the large tables require 4 times the amount of wood as the small chairs. He only has enough of a special glazing compound to glaze 16 of the small chairs or 8 of the large tables. Let X1 the number of small chairs and X2 the number of large tables. The smaller chairs sell for $3 each, while the larger tables would bring $9 each.
(a) Formulate the Problem.
(b) Solve the Linear Programming Problem.
(c) Solve also graphically
Answer:
(a) in the step-by-step explanation
(b) The optimal solution is 8 chairs and 4 tables.
(c) Graph attached
Step-by-step explanation:
(a)
C: number of small chairs
T: number of large tables
Maximize Income = 9T + 3C
Restrictions:
Wood: 4T+C<=24
Glazing: 2T+C<=16
In the graph its painted in green the "feasible region" where lies every solutions that fit the restrictions.
One of the three points marked in the graph is the optimal solution.
Point 1 (C= 16, T= 0)
Income = 9*0+3*16=$ 48
Point 2 (C=8, T=4)
Income = 9*4+3*8 = $ 60
Point 3 (C=0, T=6)
Income = 9*6+3*0 = $ 54
The optimal solution is 8 chairs and 4 tables.
list the steps that you could use to solve?
x 4
— = —
3 9
Answer:
multiply by 3Step-by-step explanation:
[tex]\dfrac{x}{3}=\dfrac{4}{9} \qquad\text{has x-coefficient $\frac{1}{3}$}[/tex]
Multiply by the reciprocal of the x-coefficient. Then you have ...
[tex]x=\dfrac{4}{3}[/tex]
Express 247_10 to (a) base 7, (b) base 2, (c) base 8, and (d) base 16.
Answer:
Step-by-step explanation:
given number,
247₁₀ to be converted into
a) base 7
divide the number by 7 and write the remainder on the left side
solution is (502)₇
b) base 2
divide the number by 2 and write the remainder on the left side and write in the direction from down to up as shown in the diagram attached below.
solution is (11110111)₂
c) base 8
divide the number by 8 and write the remainder on the left side
solution is (367)₈
d) base 16
divide the number by 16 and write the remainder on the left side
solution is (F 7)₈
15 - F
diagram is attached below.
R1 R3 in an electric instrument called a Wheatstone bridge electric resistances are related byR1/R2=R3/R4. Find R 2 if R1 = 10.00 Ω R3-6470, and R2 = R4-15.0 . If necessary, round to two decimal places.
Answer:
R2 = 43.03 ohms
Step-by-step explanation:
If R2=R4-15, then R4 = R2+15
According to the Wheastone bridge equation we have:
[tex]\frac{R1}{R2} =\frac{R3}{R4}\\\frac{10}{R2} =\frac{6470}{R2+15}\\\\10*(R2+15) = 6470*R2\\10*R2+150 = 6470*R2\\150=(6470-15)*R2\\R2=\frac{6455}{150}= 43.03333[/tex]
Approximately 0.02% of a 100-mg dose of the drug miglitol (Glyset) has been shown to appear in human breast milk. Calculate the quantity of drug detected, in milligrams, following a single dose.
Approximately 0.02 mg of the drug miglitol (Glyset) would be detected in human breast milk.
Given that, approximately 0.02% of a 100-mg dose appears in human breast milk.
To calculate the quantity of drug detected in milligrams following a single dose, we can use the given information that approximately 0.02% of a 100-mg dose appears in human breast milk.
Step-by-step calculation:
1. Convert 0.02% to a decimal by dividing it by 100: 0.02/100 = 0.0002.
2. Multiply the decimal by the dose of the drug: 0.0002 * 100 mg = 0.02 mg.
Therefore, following a single dose, approximately 0.02 mg of the drug miglitol (Glyset) would be detected in human breast milk.
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- Meredith picked 4 times as many green
peppers as red peppers. If she picked a total
of 20 peppers, how many green peppers did
she pick?
Answer:
16.
Step-by-step explanation:
The ratio is 4:1 so 4 / (4 + 1) = 4/5 of the total is green peppers.
So it is 20 * 4/5 = 16 .
In compounding a prescription, a pharmacist weighed 0.050 g of a substance on a balance insensitive to quantities smaller than 0.004 g. What was the maximum potential error in terms of percentage?
The maximum potential error is 92% as per the concept of percentage.
The pharmacist weighed 0.050 g of a substance on a balance insensitive to quantities smaller than 0.004 g.
To find the maximum potential error in terms of percentage, we need to determine the difference between the actual weight of the substance and the closest value that the balance can measure, which is 0.004 g.
The difference is 0.050 g - 0.004 g = 0.046 g.
The maximum potential error is the difference between the actual weight and the closest value that the balance can measure, divided by the actual weight, multiplied by 100%.
Therefore, the maximum potential error in terms of percentage is (0.046 g / 0.050 g) x 100% = 92%.
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The question deals with the calculation of the maximum potential error in a measurement. Given the insensitivity of the balance to 0.004 g and the actual measurement of substance of 0.050 g, the maximum potential error by calculation comes out to be 8%.
Explanation:The question is asking about the potential error in a measurement made by a pharmacist. The error is the difference between the smallest measurable quantity by the balance and the actual measurement. In this case, we have a balance that is insensitive to quantities smaller than 0.004 g, and the pharmacist is measuring 0.050 g of a substance.
To find the potential error percentage, we take the maximum potential error (which is defined by the sensitivity of the balance, 0.004 g), divide it by the actual measurement (0.050 g) and multiply by 100 to make it a percentage.
Maximum potential error percentage = (0.004 g / 0.050 g) * 100% = 8%
So the maximum potential error in this measurement is 8%.
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Consider the differential equation y'+\lambda y=e^{-t}, when \lambda is some constant.
(a) Find all values of \lambda such that all solutions tend to zero as t \rightarrow infinity .
(b) At least one solution goes to zero as t \rightarrow infinity .
Answer:
Part a) value of [tex]\lambda [/tex] such that all the solutions tend to zero equals 1.
Part b)
For a particular solution to tend to 0 will depend on the boundary conditions.
Step-by-step explanation:
The given differential equation is
[tex]y'+\lambda y=e^{-t}[/tex]
This is a linear differential equation of first order of form [tex]\frac{dy}{dt}+P(t)\cdot y=Q(t)[/tex] whose solution is given by
[tex]y\cdot e^{\int P(t)dt}=\int e^{\int P(t)dt}\cdot Q(t)dt[/tex]
Applying values we get
[tex]y\cdot e^{\int \lambda dt}=\int e^{\int \lambda dt}\cdot e^{-t}dt\\\\y\cdot e^{\lambda t}=\int (e^{(\lambda -1)t})dt\\\\y\cdot e^{\lambda t}=\frac{e^{(\lambda -1)t}}{(\lambda -1)}+c\\\\\therefore y(t)=\frac{c_{1}}{\lambda -1}(e^{-t}+c_{2}e^{-\lambda t})[/tex]
here [tex]c_{1},c_{2}[/tex] are arbitrary constants
part 1)
For all the function to approach 0 as t approaches infinity we have
[tex]y(t)=\lim_{t\to \infty }[\frac{c_{1}}{\lambda -1}(e^{-t}+c_{2}e^{-\lambda t})]\\\\y(\infty )=\frac{c_{1}}{\lambda -1}=0\\\\\therefore \lambda =1[/tex]
Part b)
For a particular solution to tend to 0 will depend on the boundary conditions as [tex]c_{1},c_{2}[/tex] are arbitrary constants
Suppose that the functions g and h are defined for all real numbers r as follows. gx) -4x +5 h (x) = 6x write the expressions for (g-h)(x) and (g+h)(x) and evaluate (g-h)(3). 2 o e m,曲 pe here to search
Answer: Our required values would be -10x+5, 2x+5 and -25.
Step-by-step explanation:
Since we have given that
g(x) = -4x+5
and
h(x) = 6x
We need to find (g-h)(x) and (g+h)(x).
So, (g-h)(x) is given by
[tex]g(x)-h(x)\\\\=-4x+5-6x\\\\=-10x+5[/tex]
and (g+h)(x) is given by
[tex]g(x)+h(x)\\\\=-4x+5+6x\\\\=2x+5[/tex]
and (g-h)(3) is given by
[tex]-10(3)+5\\\\=-30+5\\\\=-25[/tex]
Hence, our required values would be -10x+5, 2x+5 and -25.
Calculate: ( Round two decimal places for final answer)
1880 Milliliters(mL)=___pints(pts)
Answer:
1880 Milliliters(mL) = 3.97 pints
Step-by-step explanation:
This can be solved as a rule of three problem.
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.
Unit conversion problems, like this one, is an example of a direct relationship between measures.
Each ml has 0.002 pints. How many pints are there in 1880mL. So:
1ml - 0.002 pints
1880ml - x pints
[tex]x = 1800*0.002[/tex]
[tex]x = 3.97[/tex] pints
1880 Milliliters(mL) = 3.97 pints
Solve the linear equation: 3.4 + 2(9.7 – 4.8x) = 61.2 What are the possible steps involved in solving this equation? Check all that apply. Add 3.4 and 2. Distribute 2 to 9.7 and −4.8x. Combine 3.4 and 19.4. Divide both sides by 22.8. Subtract 22.8 from both sides. Divide both sides by −9.6.
Answer:
Distribute 2 to 9.7 and −4.8x. Combine 3.4 and 19.4. Subtract 22.8 from both sides. Divide both sides by −9.6Step-by-step explanation:
Here is the recommended solution method:
3.4 + 2(9.7 – 4.8x) = 61.2 . . . . . given
3.4 + 19.4 - 9.6x = 61.2 . . . . . . . distribute 2 to 9.7 and -4.8x
22.8 - 9.6x = 61.2 . . . . . . . . . . . . combine 3.4 and 19.4
-9.6x = 38.4 . . . . . . . . . . . . . . . . . subtract 22.8 from both sides
x = -4 . . . . . . . . . . . . . . . . . . . . . . divide both sides by -9.6
_____
Alternate solution method using different steps
You can also "undo" what is done to the variable, in reverse order. The variable has these operations performed on it:
multiply by -4.8add 9.7multiply that sum by 2add 3.4 to the productSo, another possible solution method is this:
3.4 + 2(9.7 – 4.8x) = 61.2 . . . . . given
2(9.7 -4.8x) = 57.8 . . . . . . . . . . . add the opposite of 3.4 (undo add 3.4)
9.7 -4.8x = 28.9 . . . . . . . . . . . . . divide by 2 (undo multiply by 2)
-4.8x = 19.2 . . . . . . . . . . . . . . . . . add the opposite of 9.7 (undo add 9.7)
x = -4 . . . . . . . . . . . . . . . . . . . . . . divide by -4.8 (undo multiply by -4.8)
Linear equation solutions are indeed the points where the lines or planes describing various linear equations intersect or connect. The candidate solution of a set of linear equations is indeed the collection of all feasible solution' values again for variables, and further calculation can be defined as follows:
Given:
[tex]\to \bold{3.4 + 2(9.7 -4.8x) = 61.2}\\\\[/tex]
To find:
Solve the linear equation=?
Solution:
[tex]\to \bold{3.4 + 2(9.7 -4.8x) = 61.2}\\\\\to \bold{3.4 + 19.4 -9.6x = 61.2}\\\\\to \bold{22.8 -9.6x = 61.2}\\\\\to \bold{ -9.6x = 61.2- 22.8}\\\\\to \bold{ -9.6x = 38.4}\\\\\to \bold{ x =- \frac{38.4}{9.6}}\\\\\to \bold{ x =- 4}\\\\[/tex]
Therefore, the steps are:
Distribute 2 to [tex]\bold{9.7\ and\ -4.8x}[/tex].
Combine [tex]\bold{3.4\ and \ 19.4}[/tex].
Subtract [tex]22.8[/tex] from both sides.
Divide both sides by[tex]-9.6[/tex].
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Show that n+1C = nCr-1 + nr.
Answer: The proof is given below.
Step-by-step explanation: We are given to show that the following equality is true :
[tex]^{n+1}C_r=^nC_{r-1}+^nC_r.[/tex]
We know that
the number of combinations of n different things taken r at a time is given by
[tex]^nC_r=\dfrac{n!}{r!(n-r)!}.[/tex]
Therefore, we have
[tex]R.H.S.\\\\=^nC_{r-1}+^nC_r\\\\\\=\dfrac{n!}{(r-1)!(n-(r-1))!}+\dfrac{n!}{(r)!(n-r)!}\\\\\\=\dfrac{n!}{(r-1)!(n-r+1)!}+\dfrac{n!}{(r)!(n-r)!}\\\\\\=\dfrac{n!}{(r-1)!(n-r+1)(n-r)!}+\dfrac{n!}{r(r-1)!(n-r)!}\\\\\\=\dfrac{n!}{(r-1)!(n-r)!}\left(\dfrac{1}{n-r+1}+\dfrac{1}{r}\right)\\\\\\=\dfrac{n!}{(r-1)!(n-r)!}\left(\dfrac{r+n-r+1}{(n-r+1)r}\right)\\\\\\=\dfrac{n!}{(r-1)!(n-r)!}\times\dfrac{n+1}{(n-r+1)r}\\\\\\=\dfrac{(n+1)!}{r!(n-r+1)!}\\\\\\=\dfrac{(n+1)!}{r!((n+1)-r)!}\\\\\\=^{n+1}C_r\\\\=L.H.S.[/tex]
Thus, [tex]^{n+1}C_r=^nC_{r-1}+^nC_r.[/tex]
Hence proved.
A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample. The target accuracy is a variance in measurements of 1.2 or less. If the lab technician takes 16 measurements and the variance of the measurements in the sample is 2.2, does this provide enough evidence to reject the claim that the lab technician’s accuracy is within the target accuracy? Compute the value of the appropriate test statistic
Answer and Explanation:
Given : A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample.
The target accuracy is a variance in measurements of 1.2 or less. If the lab technician takes 16 measurements and the variance of the measurements in the sample is 2.2.
To find :
1) Does this provide enough evidence to reject the claim that the lab technician’s accuracy is within the target accuracy?
2)Compute the value of the appropriate test statistic ?
Solution :
1) n=16 number of sample
The target accuracy is a variance in measurements of 1.2 or less i.e. [tex]\sigma_1^2 =1.2[/tex]
The variance of the measurements in the sample is 2.2 i.e. [tex]\sigma_2^2=2.2[/tex]
According to question,
We state the null and alternative hypotheses,
Null hypothesis [tex]H_o : \text{var}^2 \geq 1.2[/tex]
Alternative hypothesis [tex]H_a : \text{var}^2<1.2[/tex]
We claim the alternative hypothesis.
2) Compute the value of the appropriate test statistic.
Using Chi-square,
[tex]\chi =\frac{(n-1)\sigma_2^2}{\sigma_1^2}[/tex]
[tex]\chi =\frac{(16-1)(2.2)}{(1.2)}[/tex]
[tex]\chi =\frac{(15)(2.2)}{1.2}[/tex]
[tex]\chi =\frac{33}{1.2}[/tex]
[tex]\chi =27.5[/tex]
Therefore, The value of the appropriate test statistic is 27.5.
To determine if the lab technician's accuracy is within the target accuracy, a one-sample variance test can be performed using the chi-square statistic.
Explanation:To determine if the lab technician's accuracy is within the target accuracy, we can perform a hypothesis test. We will use a one-sample variance test to compare the sample variance to the target variance.
The appropriate test statistic for a one-sample variance test is the chi-square statistic. The chi-square statistic is calculated by taking the sample variance and dividing it by the target variance, then multiplying by the degrees of freedom.
In this case, we have 16 measurements and a target variance of 1.2. The sample variance is 2.2. We can calculate the test statistic using the formula chi-square = (n-1) * (sample variance / target variance). Plugging in the values, we get chi-square = (16-1) * (2.2 / 1.2) = 29.67.
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Estimate the sum. Use benchmarks with decimal parts of 0, 0.25, 0.50, or 0.75.
8.94+0.72
A. 9.25
B. 9.50
C. 9.75
Answer:
the estimated answer is A: 9.75.
(the actual answer is 9.66, so rounding up makes it 9.75)
Harper has $15to spend at the grocery store.She is going to buy bags of fruit that cost $4.75 each and one box of crackers that costs$3.50.Write and solve an inequality that models this situation and could be used to determine the maximum number of bags of fruit Harper can buy
Answer:
The maximum number of fruits bag Harper can buy are 3.
Step-by-step explanation:
Let there be x bags of fruits.
Let there be y boxes of chocolates
Cost of 1 bag of fruit = $4.75
So, cost of x bags = $4.75x
Cost of one box of crackers costs = $3.50
As per the given situation, the inequality forms:
[tex]4.75x+3.50y\leq 15[/tex]
So, the maximum number of bags of fruit Harper can buy, is when she buys no box of cracker.
So, putting y = 0 in above inequality , we have,
[tex]4.75x+3.50(0)\leq 15[/tex]
=> [tex]4.75x+0\leq 15[/tex]
=> [tex]4.75x \leq 15[/tex]
[tex]x\leq 3.15[/tex] rounding to 3.
Hence, the maximum number of fruits bag Harper can buy are 3.
A small business owner converts 120 J of her internal energy to electricity by peddling a bike for 1 second, which for a conversion efficiency of 40 percent, amounts to a power output of 48 watts. if she peddles a bike for 8 hours each day about how much money can she in one day assuming she sells for the electricity for 10.43 cents/ kWh
Answer:
4.00512 cents
Step-by-step explanation:
Given:
Power output = 48 Watts
Time for which owner paddles = 8 hours
Selling price of the electricity = 10.43 cents/kWh
Now,
Power = Energy × Time
or
Power generated = 48 × 8 = 384 Wh = 0.384 kWh
now,
Money earned will be = Power generated × selling price per kWh
or
Money earned = 0.384 kWh × 10.43 cents/ kWh = 4.00512 cents
Which expressions are equal to 11x10^3? Select all that apply.
A. 11x100
B. 11,000
C. 11x10x10x10
D. 33,000
E. 11x11x11
The required simplified expression for 11 x 10³ is 11,000. Option B is correct.
Given that,
A simplified expression form of the expression 11 x 10³, is to be determined.
The process in mathematics to operate and interpret the function or expression to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
= 11 x 10³
Since 10³ = 10 x 10 x 10 = 1000
= 11 x 1000
= 11,000
Thus, the required simplified expression for 11 x 10³ is 11,000. Option B is correct.
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use the binomial theorem to expand the expression :
(3x + y)^5 and simplify.
(b) find the middle term in the expansion of
(1/x+√x)^4 and simplify your unswer.
(c) determine the coefficient of x^11 in the expansion of (x^2 +1/x)^10, simplify your answer.
Answer:
a) [tex](3x+y)^5=243x^5+405x^4y+270x^3y^2+90x^2y^3+15xy^4+y^5[/tex].
b) The middle term in the expansion is [tex]\frac{6}{x}[/tex].
c) The coefficient of [tex]x^{11}[/tex] is 120.
Step-by-step explanation:
Remember that the binomial theorem say that [tex](x+y)^n=\sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^{k}[/tex]
a) [tex](3x+y)^5=\sum_{k=0}^5\binom{5}{k}3^{n-k}x^{n-k}y^k[/tex]
Expanding we have that
[tex]\binom{5}{0}3^5x^5+\binom{5}{1}3^4x^4y+\binom{5}{2}3^3x^3y^2+\binom{5}{3}3^2x^2y^3+\binom{5}{4}3xy^4+\binom{5}{5}y^5[/tex]
symplifying,
[tex](3x+y)^5=243x^5+405x^4y+270x^3y^2+90x^2y^3+15xy^4+y^5[/tex].
b) The middle term in the expansion of [tex](\frac{1}{x} +\sqrt{x})^4=\sum_{k=0}^{4}\binom{4}{k}\frac{1}{x^{4-k}}x^{\frac{k}{2}}[/tex] correspond to k=2. Then [tex]\binom{4}{2}\frac{1}{x^2}x^{\frac{2}{2}}=\frac{6}{x}[/tex].
c) [tex](x^2+\frac{1}{x})^{10}=\sum_{k=0}^{10}\binom{10}{k}x^{2(10-k)}\frac{1}{x^k}=\sum_{k=0}^{10}\binom{10}{k}x^{20-2k}\frac{1}{x^k}=\sum_{k=0}^{10}\binom{10}{k}x^{20-3k}[/tex]
Since we need that 11=20-3k, then k=3.
Then the coefficient of [tex]x^{11}[/tex] is [tex]\binom{10}{3}=120[/tex]
Indicate which of the following are propositions (assume that x and y are real numbers).
a) The integer 36 is even.
b) Is the integer 315 − 8 even?
c) The product of 3 and 4 is 11.
d) The sum of x and y is 12.
e) If x > 2, then x 2 > 3.
f) 52 − 5 + 3
Answer:
a) It is a proposition .
b) It is not a proposition.
c) It is a proposition.
d) It is a proposition.
e) It is a proposition.
f) It is not a proposition.
Step-by-step explanation:
a) The integer 36 is even: It is a proposition, since this statement can be assigned a true value. If 36 is an even number, the statement is true, but if 36 is an odd number, the statement is false.
b) Is the integer 315 - 8 even ?: It is not a proposition, since this question cannot be assigned a true value.
c) The product of 3 and 4 is 11: It is a proposition, since this statement can be assigned a true value. If 3x4 = 11, the statement is true, but if 3x4 is not 11, the statement is false.
d) The sum of x and y is 12: It is a proposition, since, this statement can be assigned a true value. If x + y = 12, the statement is true, but if x + y is not 12, the statement is false.
e) If x> 2, then x 2> 3: It is a proposition, since, this statement can be assigned a truth value.
f) 52 - 5 + 3: It is not a proposition, since this statement cannot be assigned a true value.
Beginning with Newton's second law of motion, derive the equations of motion for a projectile fired from altitude h above the ground at an angle e to the horizontal and with initial speed equal to vo.
Answer:
Considering the fire point at (0,h), x-direction positive to the right (→) and y-direction positive to up (↑) and the only force acting after fire is the projectile weight = -mg in the y-direction.
[tex]\\ x(t)=Vo*cos(e)*t\\ v_x(t)=Vo*cos(e)\\ a_y(t)=0\\ y(t)=h+Vo*sin(e)*t-\frac{g}{2}t^{2}\\ v_x(t)=Vo*sin(e)-gt\\ a_y(t)=-g[/tex]
Step-by-step explanation:
First, we apply the Second Newton's Law in both x and y directions:
x-direction:
[tex]\sum F_x= m\frac{dv_x}{dt} =0[/tex]
Integrating we have
[tex]\int\limits^{V_x} _{V_{0x}}{}\, dV_x =\int\limits^{t} _0{0}\, dt\\ V_{0x}=Vo*cos(e)\\ V_x(t)=Vo*cos(e)[/tex]
Taking into account that a=(dv/dt) and v=(dx/dt):
[tex]a_x(t)=\frac{dV_x(t)}{dt}=0\\V_x(t)=\frac{dx(t)}{dt}-->\int\limits^x_0 {} dx = \int\limits^t_0 {Vo*cos(t)} \, dt \\x(t)=Vo*cos(e)*t[/tex]
y-direction:
[tex]\sum F_y= m\frac{dv_x}{dt} =-mg[/tex]
Integrating we have
[tex]\int\limits^{V_y} _{V_{0y}}{}\, dV_y =\int\limits^{t} _0 {-g} \, dt\\ V_{0y}=Vo*sin(e)\\ V_y(t)=Vo*sin(e)-g*t[/tex]
Taking into account that a=(dv/dt) and v=(dy/dt):
[tex]a_y(t)=\frac{dV_y(t)}{dt}=-g\\V_y(t)=\frac{dy(t)}{dt}-->\int\limits^y_h {} dy = \int\limits^t_0 {(Vo*sin(t)-g*t)} \, dt \\y(t)=h+Vo*sin(e)*t-\frac{g}{2}t^{2}[/tex]