You deposit $27,580.00 in an interest-bearing account. After one year, your accrued interest is $1,442.43.

What is the interest rate of this account?

Principal:
$
27
,
580.00
time:
1
year

Interest:
$
1
,
442.43
r

=

I
P
t


Round your answer to the nearest hundredth of a percent, if necessary.

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.

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]

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.

Answer:

Step-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:

So, 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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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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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

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.

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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Answer:

The fist four terms are 0, -1, -4, -6.

Step-by-step explanation:

Since [tex] k\geq 0 [/tex], the fist four terms of the sequence correspond to those in wich [tex] k [/tex] takes the values 0, 1, 2 and 3, that is

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.

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.

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]

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.

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 .

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.

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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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.

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.

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

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]

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.

Answer:

the estimated answer is A: 9.75.

(the actual answer is 9.66, so rounding up makes it 9.75)

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]

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.

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

Answer:

The amount of principal Jamie borrowed was $3600.

Step-by-step explanation:

Let the principal of loan is = Po

[tex]P=Po\times0.12\times0.5[/tex]

Here r = 12% or 0.12

t = 6 months or 0.5 years

Now as given : Po+Interest=3816

[tex]Po+0.06Po=3816[/tex]

[tex]=>1.06Po=3816[/tex]

So, Po=3600

Hence, the amount of principal Jamie borrowed was $3600.

The average speed for this additional distance is 6.57 mi/h

The given parameters include;

To find:

The 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.

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.

Answer:

Step-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]

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.

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.

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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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