Answer:
r=1 and t= -21/2.
Step-by-step explanation:
Two vectors are parallel if both are multiples. That is, for a vector (x,y), the parallel vector to (x,y) will be of the form k(x,y) with k a real number. Then,
a) (8r, -2) = 2(4r,-1). Then, we need to have that r=1, in other case the first component wouldn't be 4 or the second component wouldn't be -1 and the vector (8r,-2) wouldn't be parallel to (4, -1).
b) for the case of (8t, 21) we need -1 in the second component and 4 in the first component, then let t= -21/2 to factorize the -21 and get 4 in the fisrt component and -1 in the second component.
[tex](8\frac{-21}{2}, 21) = -21(\frac{8}{2}, -1) = -21(4,-1)[/tex]. In other case, the vector (8t, 21) wouldn't be parallel to (4,-1).
Vector (8r,-2) is parallel to (4,-1) when r = 1, whereas (8t, 21) cannot be made parallel to it. To determine this, we look for a scalar multiple relation between the vectors.
The question asks for what value(s) of, if any, the given vector is parallel to (4,-1). To determine if two vectors are parallel, we need to see if one is a scalar multiple of the other, which means their components in each dimension multiply by the same scalar. Let's examine the given options:
(a) (8r,-2) is parallel to (4,-1) if there exists a scalar 'k' such that 4k = 8r and -1k = -2. By solving these equations, we find that k = 2 satisfies both, meaning if r = 1, the vector is parallel to (4,-1).
(b) (8t, 21) cannot be made parallel to (4,-1) through any scalar multiplication, as there's no single scalar that would simultaneously satisfy the required equations for both components.
Therefore, vector (8r,-2) is parallel to (4,-1) for r = 1, while vector (8t, 21) cannot be parallel to it under any circumstances.
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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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.
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]
8/3(z+11)=y solve for z.
Answer:
z = (3/8)y - 11
Step-by-step explanation:
Undo what has been done to z, in reverse order. Here, 11 is added and the sum is multiplied by 8/3.
To undo the multiplication, multiply the equation by the inverse, 3/8:
(3/8)(8/3)(z+11) = (3/8)y
z +11 = (3/8)y . . . . . . . . . simplify
To undo the addition of 11, add the opposite of 11 to the equation.
z +11 -11 = (3/8)y -11
z = (3/8)y -11 . . . . . . . . . simplify
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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In a biathalon race you first ride a bicycle at an average speed of 21.8 mi/h for 16.5 miles, then you must run for another 5.5 miles. With what average speed, in miles per hour, must you run if your average speed for the entire race is to be 13.8 mi/h?
The average speed for this additional distance is 6.57 mi/h
The given parameters include;
the average speed on bicycle = 21.8 mi/hthe distance traveled on bicycle = 16.5 milesadditional distance to be covered = 5.5 mileslet the average speed for this additional distance = vthe average speed for the entire race = 13.8 mi/hTo find:
the average speed for this additional distance, vThe average speed formula is given as;
[tex]average\ speed = \frac{total \ distance }{total \ time} \\\\average\ speed = \frac{16.5 \ miles \ +\ 5.5 \ miles}{\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} } \\\\13.8 \ mi/h = \frac{(16.5 \ miles \ +\ 5.5 \ miles)}{(\frac{16.5 \ mi}{21.8 \ mi/h}\ \ +\ \ \frac{5.5 \ mi}{v} ) }\\\\13.8 = \frac{(16.5 \ +\ 5.5 )}{(\frac{16.5 }{21.8 }\ \ +\ \ \frac{5.5 }{v} ) }\\\\13.8 = \frac{22 }{0.757\ \ +\ \ \frac{5.5 }{v} }\\\\13.8 (0.757\ \ +\ \ \frac{5.5 }{v} ) = 22\\\\[/tex]
[tex]10.446 + \frac{75.9}{v} = 22\\\\\frac{75.9}{v} = 22-10.446\\\\\frac{75.9}{v} = 11.554\\\\v = \frac{75.9}{11.554} \\\\v = 6.57 \ mi/h[/tex]
Thus, the average speed for this additional distance is 6.57 mi/h
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The runner must run at an average speed of 6.57 mi/h to achieve the overall average speed of 13.8 mi/h for the entire biathlon race.
Explanation:To solve for the average running speed needed to achieve an overall average speed of 13.8 mi/h in the biathlon, we will use the concept of weighted averages. The total time for both cycling and running can be expressed in terms of the distances traveled and the speeds for each segment.
First, we calculate the time taken to complete the cycling portion, which is 16.5 miles at an average speed of 21.8 mi/h. The formula for time is distance divided by speed:
Next, we need to find the running speed. Let's call the average running speed 'R'. Using the same formula for time:
The overall average speed for the entire race is the total distance divided by the total time. The total distance is 16.5 miles + 5.5 miles = 22 miles. Let 'T' represent the total time for the race.
Average speed for the entire race = Total distance / Total time = 22 miles / T
Given that the overall average speed must be 13.8 mi/h, we set up the equation:
13.8 mi/h = 22 miles / T
Now we can express 'T' as the sum of the cycling time and running time:
T = Time (cycling) + Time (running) = 0.7565 hours + (5.5 miles / R)
Substitute this into our average speed equation:
13.8 mi/h = 22 miles / (0.7565 hours + (5.5 miles / R))
Now, we solve for 'R', which represents the average running speed. We cross multiply and isolate 'R' to obtain:
(13.8 mi/h) (0.7565 hours + (5.5 miles / R)) = 22 miles
This simplifies to:
10.4469 hours + 75.9 miles / R = 22 miles
Subtract 10.4469 hours from both sides:
75.9 miles / R = 11.5531 miles
Finally, divide 75.9 by 11.5531 to find the average running speed 'R':
R = 6.57 mi/h
The runner must run at an average speed of 6.57 mi/h to have an overall average speed of 13.8 mi/h for the biathlon.
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.
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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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
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
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]
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.
If f(x) = 2x + 6 and g(x) = x, what is (gºf)(0)?
Final answer:
To find (g°f)(0), evaluate f(0) to get 6, then apply g to this result to also get 6. Hence, (g°f)(0) equals 6.
Explanation:
The question is asking to evaluate the composition of two functions, (g°f)(x), at the value x = 0. Composition of functions involves applying one function to the result of another. In this case, since f(x) = 2x + 6 and g(x) = x, we first find f(0), which is the result of plugging 0 into f(x).
Let's calculate:
f(0) = 2(0) + 6 = 6.
Next, we apply g to this result:
g(f(0)) = g(6) = 6.
Therefore, (g°f)(0) = 6.
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.
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}.
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
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)
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.
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.
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.
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.
- 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 .
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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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.
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]
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]
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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The height, h, of a ball that is tossed into the air is a function of the time, t, it is in the air. The height in feet for t seconds is given by the function h(t)=−16t2+96t
What is the domain of the function?
Question 8 options:
a)
[0, [infinity] )
b)
(-[infinity], [infinity])
c)
(0, [infinity])
d)
(0, 5)
e)
none
Answer:
e) none
Step-by-step explanation:
The first thing that we need to realise is that this question is set to a practical example of a ball being tossed into the air for a period of time. Thus, we know that:
a) time cannot be negative
b) a ball cannot travel a negative distance in height from where it was tossed it into the air (given that this starting position is also the ground)
In the context of what the problem asks us to achieve, which is to find the domain, we now know that:
a) t ≥ 0
b) the function h(t) is only practical when h(t) ≥ 0
In other words, we know that the domain starts at t = 0, however we need to find when the ball drops back onto the ground. We can do this by solving h(t) = 0:
h(t) = -16t^(2) + 96t
0 = -16t^(2) + 96t
0 = -16t(t - 6) (Factorise -16t^(2) + 96t)
So, now we get:
-16t = 0, therefor t = 0
or
t - 6 = 0, therefor t = 6
Thus, the ball is on the ground at t = 0 seconds and t = 6 seconds; it is between these two values of t that the domain exists and that the problem is practical.
Now, looking at the multiple choice options, it seems as though none of them are correct, therefor the answer would be e) none.
The other way to work through this question (particularly if it is just multiple choice) is that, after realising that the ball is tossed into the air at t = 0 seconds and then drops at some point later, we could already discount a), b) and c) as answers since they include infinity in the domain, which is not practical for this problem.
Then, we could substitute t = 5 into the equation to get:
h(t) = -16(5)^2 + 96*5
h(t) = -16*5*5 + 96*5
h(t) = -80*5 + 96*5
Since we need h(t) to be 0 at the end value of the domain, this is not the correct answer. Thus, the answer is e) none.
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]
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.