which statement best describes the interquartile range of set of weights

120 115 135 105 80 160

Answer:

Required conclusion is that if [tex]y_1, y_2[/tex]  satisfies given differential equation and wronskean is zero then they are considered as solution of that differential equation.

Step-by-step explanation:

Given differential equation,

[tex]t^2y''+3ty'+y=0[/tex] [tex] t>0\hfill (1)[/tex]

(i) To verify [tex]y_1(t)=t[/tex] is a solution or not we have to show,

[tex]t^2y_{1}^{''}+3ty_{1}^{'}+y_1=0[/tex]

But,

[tex]t^2y_{1}^{''}+3ty_{1}^{'}+y_1=(t^2\times 0)=(3t\times 1)+t=4t\neq 0[/tex]

hence [tex]y_1[/tex] is not a solution of (1).

Now if [tex]y_2=t-1[/tex] is another solution where [tex]y_2(t)=t-1[/tex] then,

[tex]t^2y_{2}^{''}+3ty_{2}^{'}+y_2=0[/tex]

But,

[tex]t^2y_{2}^{''}+3ty_{2}^{'}+y_2=(t^2\times 0)+(3t\times 1)+t-1=4t-1\neq 0[/tex]

so [tex]y_2[/tex] is not a solution of (1).

(ii) Rather the wronskean,

[tex]W(y_1,y_2)=y_{1}y_{2}^{'}-y_{2}y_{1}^{'}=(t\times 1)-((t-1)\times 1)=t-t+1=1\neq 0[/tex]

Hence it is conclude that if [tex]y_1, y_2[/tex] satisfies (i) along with condition (ii) that is wronskean zero, only then  [tex]y_1, y_2[/tex] will consider as solution of (1).

Answer:

$2,520

Step-by-step explanation:

The simple Interest earned on an deposit, P at a rate of r% for a period of t years is calculated using the formula:

[tex]\text{ Simple Interest}=\dfrac{Principal*Time*Rate}{100}[/tex]

P=$10,500

R=6%

T=4 years

Therefore:

[tex]\text{ Simple Interest}=\dfrac{10500*4*6}{100}\\=\$2,520[/tex]

Sally will earn $2520 interest after 4 years.

Answer:

  18%

Step-by-step explanation:

Of the 50 soccer players, 4 play soccer and lacrosse, and 5 play soccer and football. That is, 9 of the 50 players also play one of the other sports.

 9/50 × 100% = 18%

18% of soccer players also play another sport.

Answer:

12/7

Step-by-step explanation:

To set up a ratio of "thing 1" to "thing 2", build a fraction with "thing 1" on top (numerator) and "thing 2" on the bottom (denominator).

The ratio of in-state students to out-of-state students is [tex]\frac{108}{63}[/tex] and this can be simplified ("reduced") by dividing both numbers by 9 to get the fraction 12/7.

Final answer:

Explanation on finding the ratio of students planning in-state college to out-of-state college.

Explanation:

The ratio of students planning on going to an in-state college to students planning on going to an out-of-state college:

To find the ratio, you need to divide the number of students planning in-state college by the number planning out-of-state college.

Number of students planning in-state college: 108

Number of students planning out-of-state college: 63

Ratio: 108:63 which simplifies to 36:21 or 12:7

Answer:

68% of the diameters are between 7.06 cm and 7.78 cm

Step-by-step explanation:

Mean diameter = μ = 7.42

Standard Deviation = σ = 0.36

We have to find what percentage of diameters will be between 7.06 cm and 7.78 cm. According to the empirical rule, for a bell-shaped data:

So, we first need to find how many standard deviations away are the given two data points. This can be done by converting them to z-score. A z score tells us that how far is a data value from the mean. The formula to calculate the z-score is:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

x = 7.06 converted to z score will be:

[tex]z=\frac{7.06-7.42}{0.36}=-1[/tex]

x = 7.78 converted to z score will be:

[tex]z=\frac{7.78-7.42}{0.36}=1[/tex]

This means the two given values are 1 standard deviation away from the mean and we have to find what percentage of values are within 1 standard deviation of the mean.

From the first listed point of empirical formula, we can say that 68% of the data values lie within 1 standard deviation of the mean. Therefore, 68% of the diameters are between 7.06 cm and 7.78 cm

Approximately 68% of the apples have diameters between 7.06cm and 7.78cm.

To determine the percentage of apples with diameters between 7.06cm and 7.78cm, we can use the empirical rule which is based on the standard deviation. According to the empirical rule, approximately 68% of the apples will fall within one standard deviation of the mean, which in this case is between 7.42 - 0.36 and 7.42 + 0.36. In other words, between 7.06cm and 7.78cm. Therefore, approximately 68% of the apples have diameters between 7.06cm and 7.78cm.

https://brainly.com/question/35669892

#SPJ3

Answer: the volume is: 280.8

Step-by-step explanation:

volume= length x width x height

Answer:

140.4 cm^3

Step-by-step explanation:

The formula for calculating volume of a triangular prism : 1/2 × h × l × b

1/2 × 4 × 9 × 7.2 = 140.4 cm^3

Answer:

(1,1) is a solution of the system.

Step-by-step explanation:

Let's solve the system.

y = 2x - 1

5x - 4y = 1

In the first equation, y is already separated as a function of x. So we replace in the second equation;

5x - 4(2x - 1) = 1

5x - 8x + 4 = 1

4 - 1 = 8x - 5x

x = 1

y = 2x - 1 = 2(1) - 1 = 1

(1,1) is a solution of the system.

Answer:

The ratio is 2:1

Step-by-step explanation:

This is since there are 2 times the amount of hands as there are phones

Answer:

504 ways

Step-by-step explanation:

The first position can go in 9 ways

The second position can go to the team in 8 ways

The third position can go to the team in 7 ways

Therefore, the first,second and third position can go to the team in:

9 X 8 X 7=504 ways

or Simply, we can use Permutation.

[tex]^9P_3=504[/tex] ways

Answer:

See below.

Step-by-step explanation:

A year has 12 months and 365 days.

3 months: 3/12 year = 1/4 year

55 days: 55/365 year = 11/73 year

1 month: 1/12 year

7 months = 7/12 year

120 days = 120/365 year = 24/73 year

Answer:

3 months = 3/12 | 55 days = 55/365 | 1 month = 1/12 | 7 months = 7/12 | 120 days = 120/365

Step-by-step explanation:

All you have to do is write the numbers with the total underneath. With months, it is out of 12 because there are 12 months in a year. With days, it is out of 365 days because there are 365 days in total in a year.

If needed, simplify the answer to its simplest form

Answer:

[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

1. Let's draw the trapezoids, then combine them. The first trapezoid has larger Base measuring 4.67 cm, parallel and minor base =2, an area of  4.98

2. Since the other one is a copy, same area, same base. The junction of both trapezoids generates a hexagon. We have another trapezoid with an area of 4.98. The hexagon has a total area of 9.96

3. So each trapezoid has exactly 1/2 of the area of the hexagon.

Using a system of equations, it was determined there were 336 bottles worth 5 cents each and 82 bottles worth 10 cents each, summing up to a total of $25.

The local organization collected 418 returnable bottles and cans, some worth 5 cents each and the rest worth 10 cents each. The total value was $25.00. We need to find out how many bottles were worth 5 cents and how many were worth 10 cents.

We can set up two equations to solve this problem using algebra. Let's assume that the number of 5-cent bottles is x and the number of 10-cent bottles is y.

The first equation will represent the total number of bottles: x + y = 418.

The second equation will represent the total value of these bottles: 0.05x + 0.10y = 25.

To solve this system of equations, we can multiply the second equation by 100 to eliminate the decimals and then apply the method of substitution or elimination to find the values of x and y.

Rewrite the second equation without decimals: 5x + 10y = 2500.

Multiply the first equation by 5: 5x + 5y = 2090.

Subtract the modified first equation from the second equation: 5y = 410.

Divide both sides by 5 to find y: y = 82.

Substitute y = 82 into the first equation and solve for x: x = 418 - 82 = 336.

Therefore, there were 336 bottles or cans worth 5 cents each and 82 bottles or cans worth 10 cents each.

Answer:

a. 0.1576<p<0.2310

b. The two restaurants likely have similar order rates which are inaccurate.

Step-by-step explanation:

a. We first calculate the proportion, [tex]\hat p[/tex]:

[tex]\hat p=\frac{61}{314}\\\\=0.1943[/tex]

-We use the z-value alongside the proportion to calculate the margin of error:

[tex]MOE=z\sqrt{\frac{\hat p(1-\hat p)}{n}}\\\\=1.645\times \sqrt{\frac{0.1943(1-0.1943)}{314}}\\\\=0.0367[/tex]

The confidence interval at 90% is then calculated as:

[tex]CI=\hat p\pm MOE\\\\=0.1943\pm 0.0367\\\\=[0.1576,0.2310][/tex]

Hence, the confidence interval at 90% is [0.1576,0.2310]

b. From a above, the calculated confidence interval is 0.1576<p<0.2310

-We compare the calculated CI to the stated CI of 0.147<p<0.206

-The two confidence intervals overlap each other and have the same value for 0.1576<p<0.206

-Hence, we conclude that the two restaurants likely have similar order rates which are inaccurate.

Answer:

Zinc Phosphate cement

Answer:

luting agent

Step-by-step explanation:

For which of the following procedures would you include a temporary

luting agent is basicslly the same thing

0.555555556 so that's it hope this helped

Answer:

0.5

Step-by-step explanation:

Because I said so

Answer:

2(4x− 9) = 5(x − 4)

=>8x- 18= 5x-20

=>8x-5x= -20+18

=>3x = -2

=>x= -2/3

Answer:

4, -4

Step-by-step explanation:

Take the  (+2)th  root of both sides of the  equation  to eliminate the exponent on the left side.

The complete solution is the result of both the positive and negative portions of the solution.

x  =  4 , − 4

Answer:

choosing 1 red marble ; choosing 1 white marble, replacing it, and choosing another white marble ; and choosing 1 white marble

Step-by-step explanation:

There are 2+3+5 = 10 marbles.  The probability of choosing 1 blue marble is 2/10 = 1/5; this is not greater than 1/5.

The probability of choosing 1 red marble is 3/10; this is greater than 2/10, which is the same as 1/5.

The probability of choosing 1 red marble is 3/10; not replacing it and choosing a blue marble would then be a probability of 2/9.  Together this is a probability of 3/10(2/9) = 6/90 = 3/45; this is smaller than 9/45, which is the same as 1/5.

The probability of choosing 1 white marble is 5/10 = 1/2; replacing it and choosing another white marble would be 1/2.  Together this is a probability of 1/2(1/2) = 1/4; this is greater than 4/20, which is the same as 1/5.

The probability of choosing 1 white marble is 5/10 = 1/2.  This is greater than 2/10, which is the same as 1/5.

Answer:

b,d,e are the awnsers

Step-by-step explanation:

Answer:

0.0498 = 4.98%

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

In which

x is the number of sucesses

e = 2.71828 is the Euler number

[tex]\mu[/tex] is the mean in the given time interval.

Inquiries arrive at a record message device according to a Poisson process of rate 15 inquiries per minute.

Each minute has 60 seconds.

So a rate of 1 inquire each 4 seconds.

The probability that it takes more than 12 seconds for the first inquiry to arrive is approximately

Mean of 1 inquire each 4 seconds, so for 12 seconds [tex]\mu = \frac{12}{4} = 3[/tex]

This probability is P(X = 0).

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498[/tex]

The probability that it takes more than 12 seconds for the first inquiry to arrive in a Poisson process at a rate of 15 inquiries per minute is calculated using the exponential distribution formula: e^{-15*0.2}.

The student is asking for the probability that it takes more than 12 seconds for the first inquiry to arrive at a record message device, with inquiries arriving according to a Poisson process with a rate of 15 inquiries per minute. Since the time between arrivals in a Poisson process follows an exponential distribution, we can calculate this probability using the exponential distribution's formula.

The mean interval between inquiries is the inverse of the rate, which for 15 inquiries per minute is 1/15 minute per inquiry, or 4 seconds per inquiry. To convert minutes to seconds, multiply by 60. Therefore, the average interval is 4 seconds.

The exponential distribution gives us the probability that the time until the first event (inquiry) exceeds a certain amount, t, which is P(T > t) = e-λ*t, where λ is the rate and T is the time. In this case, t is 12 seconds or 0.2 minutes. Therefore, the probability (P) that the time is more than 12 seconds is: P(T > 0.2) = e-15*0.2.

Answer:

57 in³

Step-by-step explanation:

Area of cylinder= [tex]\pi {r}^{2} h[/tex]

Let's find the radius.

Radius= Diameter ÷2

Radius= 6 ÷2

Radius= 3 in.

Area of Rover's dog bowl

= 3.14(3)²(2)

= 56.52

= 57 in³ (nearest cubic inch)

Answer:

57 in3

Step-by-step explanation:

Answer:

The new balance is $675.8

Step-by-step explanation:

Solution

Given that

The total amount of loan = $820.

The terms were = 3/10, n/60

Within 10 days, the buyer sent in a payment of  =$140

What is the new balance =?

Now,

3/10 =  if the amount is paid in 10 days, a discount of is included

n/60 = This means that all amount should be paid within 60 days

Thus,

$140 for 3/10 as this is paid within 10 days

140 *3% = 140 * 3/100

we get

=$4.2 of discount

Then,

The balance amount becomes $ 820 - 140 -4.2

=$675.8

Answer:

THE ANSWER IS (6,-9)

Step-by-step explanation:

The solution to the equation is ( 6 , -9 )

The value of x = 6

The value of y = -9

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the first equation be represented as A

Now , the value of A is

y = x - 15   be equation (1)

Let the second equation be represented as B

Now , the value of B is

y = -2x + 3   be equation (2)

Substituting the value of equation (1) in equation (2) , we get

x - 15 = -2x + 3

On simplifying the equation , we get

Adding 2x on both sides of the equation , we get

x + 2x - 15 = 3

Adding 15 on both sides of the equation , we get

3x = 18

Divide by 3 on both sides of the equation , we get

x = 6

Therefore , the value of x is 6

Substitute the value of x in equation (1) , we get

y = x - 15

y = 6 - 15

y = -9

Therefore , the value of y is -9

Hence , the values of x and y of the equation is ( 6 , -9 )

To learn more about equations click :

https://brainly.com/question/19297665

#SPJ6

The question is incomplete! Complete question along with answer and step by step explanation is provided below.

Question:

Tommy Wait, a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his time between pitches are normally distributed with a mean of 29 seconds and a standard deviation of 2.1 seconds. What percentage of his times between pitches are longer than 31 seconds ?

Given Information:  

Mean pitching time = μ = 29 seconds

Standard deviation of pitching time = σ = 2.1 seconds

Required Information:  

P(X > 31) = ?

Answer:

[tex]P(X > 31) = 17.11 \%[/tex]

Step-by-step explanation:

What is Normal Distribution?

We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.  

We want to find out the probability that what percentage of his times between pitches are longer than 31 seconds.

[tex]P(X > 31) = 1 - P(X < 31)\\P(X > 31) = 1 - P(Z < \frac{x - \mu}{\sigma} )\\P(X > 31) = 1 - P(Z < \frac{31 - 29}{2.1} )\\P(X > 31) = 1 - P(Z < \frac{2}{2.1} )\\P(X > 31) = 1 - P(Z < 0.95)\\[/tex]

The z-score corresponding to 0.95 is 0.8289

[tex]P(X > 31) = 1 - 0.8289\\P(X > 31) = 0.1711\\P(X > 31) = 17.11 \%[/tex]

Therefore, 17.11% of his times between pitches are longer than 31 seconds.

How to use z-table?

Step 1:

In the z-table, find the two-digit number on the left side corresponding to your z-score. (e.g 0.9 1.4, 2.2, 0.5 etc.)

Step 2:

Then look up at the top of z-table to find the remaining decimal point in the range of 0.00 to 0.09. (e.g. if you are looking for 0.95 then go for 0.05 column)

Step 3:

Finally, find the corresponding probability from the z-table at the intersection of step 1 and step 2.

To solve for p in the equation 1/3p + 1/2p = 7/6p + 5 + 4, you need to combine like terms and isolate p. The solution is p = 27.

To solve for p in the equation 1/3p + 1/2p = 7/6p + 5 + 4, you need to combine like terms and isolate p on one side of the equation.

Multiply the fractions by their respective denominators to eliminate the fractions. This gives us 2/6p + 3/6p = 7/6p + 9.

Combine like terms. On the left side, 2/6p + 3/6p = 5/6p. On the right side, 7/6p + 9 remains unchanged.

Subtract 5/6p from both sides to isolate p. This gives us 7/6p - 5/6p = 9.

Combine like terms on the left side. 7/6p - 5/6p = 2/6p.

Simplify the equation further. 2/6p = 9 can be reduced to 1/3p = 9.

Multiply both sides of the equation by the reciprocal of 1/3, which is 3/1. This gives us p = 9 * (3/1) = 27.

Therefore, the solution for p in the equation is p = 27.

Close off the hemisphere [tex]S[/tex] by attaching to it the disk [tex]D[/tex] of radius 3 centered at the origin in the plane [tex]z=0[/tex]. By the divergence theorem, we have

[tex]\displaystyle\iint_{S\cup D}\vec F(x,y,z)\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dV[/tex]

where [tex]R[/tex] is the interior of the joined surfaces [tex]S\cup D[/tex].

Compute the divergence of [tex]\vec F[/tex]:

[tex]\mathrm{div}\vec F(x,y,z)=\dfrac{\partial(xz^2)}{\partial x}+\dfrac{\partial\left(\frac{y^3}3+\sin z\right)}{\partial y}+\dfrac{\partial(x^2z+y^2)}{\partial k}=z^2+y^2+x^2[/tex]

Compute the integral of the divergence over [tex]R[/tex]. Easily done by converting to cylindrical or spherical coordinates. I'll do the latter:

[tex]\begin{cases}x(\rho,\theta,\varphi)=\rho\cos\theta\sin\varphi\\y(\rho,\theta,\varphi)=\rho\sin\theta\sin\varphi\\z(\rho,\theta,\varphi)=\rho\cos\varphi\end{cases}\implies\begin{cases}x^2+y^2+z^2=\rho^2\\\mathrm dV=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi\end{cases}[/tex]

So the volume integral is

[tex]\displaystyle\iiint_Rx^2+y^2+z^2\,\mathrm dV=\int_0^{\pi/2}\int_0^{2\pi}\int_0^3\rho^4\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{486\pi}5[/tex]

From this we need to subtract the contribution of

[tex]\displaystyle\iint_D\vec F(x,y,z)\cdot\mathrm d\vec S[/tex]

that is, the integral of [tex]\vec F[/tex] over the disk, oriented downward. Since [tex]z=0[/tex] in [tex]D[/tex], we have

[tex]\vec F(x,y,0)=\dfrac{y^3}3\,\vec\jmath+y^2\,\vec k[/tex]

Parameterize [tex]D[/tex] by

[tex]\vec r(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath[/tex]

where [tex]0\le u\le 3[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to be

[tex]\dfrac{\partial\vec r}{\partial v}\times\dfrac{\partial\vec r}{\partial u}=-u\,\vec k[/tex]

Then taking the dot product of [tex]\vec F[/tex] with the normal vector gives

[tex]\vec F(x(u,v),y(u,v),0)\cdot(-u\,\vec k)=-y(u,v)^2u=-u^3\sin^2v[/tex]

So the contribution of integrating [tex]\vec F[/tex] over [tex]D[/tex] is

[tex]\displaystyle\int_0^{2\pi}\int_0^3-u^3\sin^2v\,\mathrm du\,\mathrm dv=-\frac{81\pi}4[/tex]

and the value of the integral we want is

(integral of divergence of F) - (integral over D) = integral over S

==>  486π/5 - (-81π/4) = 2349π/20

The process involves applying the Divergence Theorem over a closed surface formed by adding a bottom disk to the sphere and converting the given surface integral into a volume integral which is easier to calculate. F(x, y, z) must be used accurately in all calculations.

To solve this problem, we need to apply the Divergence Theorem to evaluate the surface integral of the given vector field F(x, y, z) over the top half of the sphere. Before we do that, we must first compute the integrals over S1 and S2, where S1 is the disk x² + y² ≤ 9, oriented downward, and S2 = S1 ∪ S.

On applying the Divergence Theorem over a closed surface S₁ + S₂ obtained by adding a bottom disk to the sphere, we can convert the given surface integral into a volume integral over the region inside the closed surface.

Once we obtain this volume integral, this should simplify our calculations, as volume integrals are typically easier to evaluate than surface integrals. This strategy utilises the power of the Divergence Theorem, which connects the flow of a vector field across a surface to the behavior of the field inside the volume enclosed by the surface.

Remember to use the correct vector field formulas for F when calculating the integrals over S1 and S2. Ensure each step is carefully followed so errors are not made.

https://brainly.com/question/33180298

#SPJ3

Answer: 3

Step-by-step explanation:

E2020 answer

Answer:

3

Step-by-step explanation:

answer on edge

Step-by-step explanation:

The Length ,

L

=

284

f

t

.

Explanation:

Given:

Rectangle

Area,

A

=

8804

f

t

2

let W bet he width of the rectangle

L be the length of the rectangle

L

=

10

W

−

26

E

q

u

a

t

i

o

n

1

substitute to

e

q

u

a

t

i

o

n

2

A

=

(

L

)

(

W

)

e

q

u

a

t

i

o

n

2

A

=

(

10

W

−

26

)

(

W

)

8804

=

(

10

W

−

26

)

(

W

)

factor

8804

=

2

(

5

W

−

13

)

(

W

)

divide both sides by 2

4402

=

(

5

W

−

13

)

(

W

)

4402

=

5

W

2

−

13

W

transposing 4402 to the right side of the equation

0

=

5

W

2

−

13

W

−

4402

by quadratic formula

W

=

−

(

−

13

)

+

√

(

−

13

)

2

−

4

(

5

)

(

−

4402

)

2

(

5

)

W

=

[

13

+

√

169

+

88040

]

10

W

=

13

+

(

√

88209

)

10

W

=

13

+

297

10

W

=

310

10

W

=

31

ft

Thus ,

L

=

10

W

−

26

=

10

(

31

)

−

26

L

=

284

f

t

.

answer

W

=

−

(

−

13

)

−

√

−

(

−

13

2

)

−

4

(

5

)

(

−

4402

)

2

(

5

)

this is discarded since this will yield a negative

To find the length of the rectangle, set up an equation using the given information. Solve the quadratic equation to find the width and substitute it back to find the length.

To find the length of the rectangle, we can set up an equation using the given information. Let's assume the width of the rectangle is 'w' meters. According to the problem, the length of the rectangle is 10 times its width minus 11 meters, so it can be represented as 10w - 11 meters. The area of the rectangle is given as 9888 square meters. We know that the formula for the area of a rectangle is length times width, so we can write the equation as:

w (10w - 11) = 9888

Expanding the equation and rearranging terms, we get:

10w^2 - 11w - 9888 = 0

Now, we can solve this quadratic equation for 'w' and find the width of the rectangle. Once we have the width, we can substitute it back into the expression for the length (10w - 11) to find the length of the rectangle.

https://brainly.com/question/34224051

#SPJ2

Final answer:

To simplify √72 by removing all perfect squares, you factor 72 into 2³ × 3², then take out the square root of 4 and 9 to get 6√(2) as the simplified result.

Explanation:

To simplify the square root of 72 and remove all perfect squares from inside the square root, you first need to factor 72 into its prime factors. The prime factorization of 72 is 2³ × 3². This can be rewritten as (2² × 3²) × 2, which simplifies to (4 × 9) × 2.

Since the square root of 4 and 9 are perfect squares, you can take them out of the square root, resulting in √(4) × √(9) × √(2) = 2 × 3 × √(2) = 6√(2).

Therefore, the simplified form of √72 is 6√(2), which removes all perfect squares from inside the square root.

Final answer:

To find the "long" side opposite the 60° angle in a 30-60-90 triangle with a "short" side length of 7, we multiply 7 by √3 to get 7√3.

Explanation:

The student appears to be asking about the relationships between the sides of a 30-60-90 triangle, which is a special type of right triangle. In such a triangle, the sides are in the ratio 1:√3:2. So, if the "short" side opposite the 30° angle is given as 7, then the "long" side opposite the 60° angle can be found by multiplying the short side by √3. To find the hypotenuse (opposite the 90° angle), we would multiply the short side by 2.

Step-by-step to find the long side:

Therefore, the length of the "long" side opposite the 60° angle is 7√3.