Rewrite the subtraction number sentence as an addition number sentence.
5- (-2)
​

The total distance from A to B is 5 ( -3 to 2 = 5).

Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.

Answer:

A to B is 5 ( -3 to 2 = 5).//!!//Then you are gonna be Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.

Final answer:

The sum of sequence P (YP) is greater than the sum of sequence Q (Q) by 94, with P summing to 2850 and Q summing to 2756.

Explanation:

To determine which sum is greater between P and Q, we need to first understand that P and Q represent the sum of arithmetic sequences. The first sequence, P, begins with 8 and increments by 2, ending at 106. The second sequence, Q, starts at 2 and increments by 2, ending at 104.

To find out the sums, we can use the formula for the sum of an arithmetic sequence: S = n/2 (a1 + an), where S is the sum, n is the number of terms, a1 is the first term, and an is the last term. For the sequence P, the first term a1 is 8, and the last term an is 106. To find n, the number of terms, we can use the formula n = (an - a1) / d + 1, where d is the common difference, which is 2 in this case.

Applying the formula to the sequence P, we get:

The number of terms in P: n(P) = (106 - 8) / 2 + 1 = 50

Sum of P: S(P) = 50/2 x (8 + 106) = 25 x 114 = 2850

For the sequence Q:

The number of terms in Q: n(Q) = (104 - 2) / 2 + 1 = 52

Sum of Q: S(Q) = 52/2 x (2 + 104) = 26 x 106 = 2756

Comparing these sums, we can see that the sum of sequence P, 2850, is greater than the sum of sequence Q, 2756. The difference between the sums is 2850 - 2756 = 94.

Therefore, P is greater than Q by 94.

Answer:

12,000.

Step-by-step explanation:

Let x be the number of people that were enrolled last year.

We have been given that enrollment at ELAC decreased by 5%, or 600 people, the year. We are asked to find the number of people that were enrolled last year.

We can set as equation such that 5% of x equals 600.

[tex]\frac{5}{100}\cdot x=600[/tex]

[tex]0.05x=600[/tex]

[tex]\frac{0.05x}{0.05}=\frac{600}{0.05}[/tex]

[tex]x=12,000[/tex]

Therefore, 12,000 people were enrolled last year.

To prove that V is a vector space we must prove that the sum define on it satisfy conmutativiy, asociativity and existence of the neutral element and inverses. Also, the scalar multiplication define on V must satisfy distributivity propertie with respect to the sum and viceversa, and an asosiativity too in the sense that [tex]x(y\cdot v)= (xy)\cdot v[/tex] for [tex]x,y\in \mathbb{R}[/tex] and [tex]v\in V[/tex]. One can prove with this that the neutral element for the sum is unique. But with your operations you  have two neutral elements for [tex](1;2)[/tex]

[tex](1;2)+(-1;3)=(0;0)[/tex]

and

[tex](1;5)+(-1;11)=(0;0)[/tex]

So, you dont have a vector space.

Final answer:

The set V, with its defined addition and scalar multiplication operations, does not fulfill essential vector space properties such as the existence of an additive identity, presence of additive inverses, and correct scalar multiplication effects on components. Therefore, V is not a vector space.

Explanation:

To determine if a set V, defined with specific addition and scalar multiplication operations, is a vector space, it must satisfy several properties commonly defined in linear algebra.

For V to be considered a vector space, the addition operation must be associative and commutative, there must be an additive identity (zero vector), each vector must have an additive inverse, scalar multiplication must be associative, there must be a multiplicative identity (1), and both operations must distribute over vector addition and scalar addition.

The defined operations on V are (x₁; y₁) + (x₂; y₂) = (x₁ + x₂; 0) for vector addition and c(x; y) = (cx; 0) for scalar multiplication. These operations fail to satisfy several vector space properties, including:

Due to these inadequacies, V does not meet the criteria for a vector space under the provided operations.

Answer: a. You must pay $11.19 for the item.

b.The price you would expect to pay would be $36.59

Step-by-step explanation:

Hi, for first the question you need to calculate the 30 percent of the price of the item, and then subtract that result to the original price.

So: $15.99 × 0,30 = $4.797

$15.99 - $4.797 = $11.19

You must pay $11.19 for the item.

It´s a similar resolution, first you calculate the 40 percent of the retail price, and then subtract that result to the retail price.

So:

$60.99 × 0,40= $24.396

$60.99 - $24.396 = $36.59

The price you would expect to pay would be $36.59

Answer:

The expended form of the provided expression is: [tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]

Step-by-step explanation:

Consider the provided expression.

[tex](2x-1)^5[/tex]

The binomial theorem:

[tex](a+b)^{n}=\sum _{r=0}^{n}{n \choose r}a^{n-r}b^r[/tex]

Where,

[tex]{n \choose r}= ^nC_r =\frac{n!}{(n-r)!r!}[/tex]

Now by using the above formula.

[tex]\frac{5!}{0!\left(5-0\right)!}\left(2x\right)^5\left(-1\right)^0+\frac{5!}{1!\left(5-1\right)!}\left(2x\right)^4\left(-1\right)^1+\frac{5!}{2!\left(5-2\right)!}\left(2x\right)^3\left(-1\right)^2+\frac{5!}{3!\left(5-3\right)!}\left(2x\right)^2\left(-1\right)^3+\frac{5!}{4!\left(5-4\right)!}\left(2x\right)^1\left(-1\right)^4+\frac{5!}{5!\left(5-5\right)!}\left(2x\right)^0\left(-1\right)^5[/tex]

[tex]2^5\cdot \:1\cdot \:1\cdot \:x^5-1\cdot \frac{2^4\cdot \:5x^4}{1!}+1\cdot \frac{2^3\cdot \:20x^3}{2!}-1\cdot \frac{2^2\cdot \:20x^2}{2!}+1\cdot \frac{5\cdot \:2x}{1!}+1\cdot \frac{\left(-1\right)^5}{\left(5-5\right)!}[/tex]

[tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]

Hence, the expended form of the provided expression is: [tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]

Final answer:

To find the length of AC in the right-angled isosceles triangle △ADC with altitude CD and AD equals to BC, we use the Pythagorean theorem, substituting known lengths to solve for AC, which is √11.25 cm.

Explanation:

The student is asking to find the length of side AC in a triangle △ABC where CD is an altitude and AD is equal to BC, with given lengths AB = 3 cm and CD = 3 cm.

To solve for AC, we can use the Pythagorean theorem which states that for a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Since CD is an altitude and AD = BC, △ADC is a right-angled isosceles triangle.

Using the Pythagorean theorem in △ADC, we have:

AD2 + CD2 = AC2

Since AD = BC and CD = 3 cm, let's assume AD = x cm. Then,

x2 + 32 = AC2

x2 + 9 = AC2

But, AB = 3 cm, and AB = AD + DB = x + x = 2x,

Therefore, x = AB/2 = 1.5 cm. Substituting this value back into the previous equation:

(1.5)2 + 9 = AC2 = 2.25 + 9 = 11.25

AC = √11.25 cm

Therefore, the length of AC is √11.25 cm.

Answer:

z=10

x1=0

x2=0

x3=3.33

Step-by-step explanation:

First Step convert your constraints in standard equations

so we have

x1 + 2x2 + 3x3+x4 = 10

x1 + x2 +x5= 5

x1 +x6= 1

Now we pass it all to the simplex table

Remember that we choose the column with the most negative value

Pivot Element=3

Divide all elements on Pivot Line by Pivot Element

Line x5= 0*Pivot Line +Line x5

Line x6= 0*Pivot Line+ Line X6

Line Z= 3* Pivot Line + Line Z

We finish when all the elements from the line Z are positive

Hence we have that x3=3.33 and x1=0, x2=0 and the max of z is 10

Answer:

z=10

x1=0

x2=0

x3=3.33

Step-by-step explanation:

Answer:

Ac = 10100100011

[tex]A \cup B = 11111111110[/tex]

[tex]A \cap B = 00010011000[/tex]

[tex]A - B = 100100111010[/tex]

Step-by-step explanation:

All these operations are bitwise operations.

Ac is the complement of a. So where we have a bit 0, the complement is 1. Where we have a bit 1, the complement is 0. So

A = 01011011100

Ac = 10100100011

The second operation is the union between A and B. This is bitwise(bit 1 of A with bit 1 of B, bit 2 of A with bit 2 of B,...). The union operation is only 0 when it is between two zeros. So:

A = 01011011100

B = 10110111010

[tex]A \cup B = 11111111110[/tex]

The third operation is the intersection between A and B. Again, bitwise. The intersection is only 1 when it is between two bits that are 1. So

A = 01011110100

B = 10110111010

[tex]A \cap B = 00010011000[/tex]

The last operation is the bitwise subtraction between A and B. We start from the least significant bit(the last one). And we have to take care of the borrow operator also, similarly to a decimal subtraction.

We can only borrow from a previous bit 1, and this bit is set to 0

0-0 is 0 with no borrow

1-0 is 1 with no borrow

1 with borrow - 0 is 1 with borrow

1 with borrow - 1 is 1 with no borrow

0-1 is 1 with no borrow

0 with borrow -1 is 0 with no borrow

1-1 is 0 with no borrow

If we arrive at the first bit(the most significant) with a borrow, we must add a 1 at the front of the answer. So

A = 01011110100

B = 10110111010

[tex]A - B = 100100111010[/tex]

Answer:  Option 'c' is correct.

Step-by-step explanation:

Since we have given that

The salary earned for a month stored in = A35

Rate of tax reduces by = 22%

We need to remove the % sign by dividing 22 by 100 and it becomes 0.22.

So, the dollar amount of taxes is given by

[tex]Taxes=A35\times \dfrac{22}{100}\\\\Taxes=A35\times 0.22[/tex]

Hence, Option 'c' is correct.

Answer:

The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.

Step-by-step explanation:

To find : How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?

Solution :

A 150-W light bulb is on for 20-hours.

The heat produced by bulb is given by,

[tex]H=150\times 20[/tex]

[tex]H=3000\ W-hr[/tex]

We know that,

[tex]1\ \text{W-hr}=3.4\ \text{BTU}[/tex]

Converting W-hr into BTU,

[tex]3000\ \text{W-hr}=3000\times 3.4\ \text{BTU}[/tex]

[tex]3000\ \text{W-hr}=10200\ \text{BTU}[/tex]

Therefore, The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.

Answer:

(a) 0.8512 (b) 0.8512 (c) 0.7024 (d) 0.0110

Step-by-step explanation:

The blood glucose follows a normal distribution N(μ=85;σ=24).

For every value of X, we can calculate the z-score (equivalent for a N(0;1)) and compute the probability.

(a) P(x>60)

z = (x-μ)/σ = (60-85)/24 = -1.0417

P(x>60) = P(z>-1.0417) = 0.8512

(b) P(x<110)

z = (x-μ)/σ = (110-85)/24 = 1.0417

P(x<110) = P(z<1.0417) = 0.8512

(c) P(60<x<110) = P(x<110)-P(x<60)

P(60<x<110) = P(z<1.0417) - P(z<-1.0417)

P(60<x<110) = 0.8512 - (1-0.8512) = 0.8512 - 0.1488 = 0.7024

(d) P(x>140)

z = (x-μ)/σ = (140-85)/24 = 2.2917

P(x>140) = P(z>2.2917) = 0.0110

Final answer:

Explanation of probabilities for different blood glucose levels using mean and standard deviation.

Explanation:

Probability calculations for blood glucose levels:

(a) x is more than 60: Calculate the z-score using the formula z = (x - μ) / σ. With x = 60, μ = 85, and σ = 24, find the probability using a standard normal distribution table.

(b) x is less than 110: Use the z-score formula with x = 110, μ = 85, and σ = 24 to determine the probability.

(c) x is between 60 and 110: Find the individual probabilities for x = 60 and x = 110, then subtract the two values to get the probability in this range.

(d) x is greater than 140: Similar to the previous steps, find the z-score for x = 140 and calculate the probability.

Answer:

[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]

Approximate error = 0.4426

Step-by-step explanation:

f(x)=tanx, a=0

Maclaurin series formula used is given below

[tex]f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)x^{n}}{n!}=f(0)+f'(0)x+\dfrac{f''(0)}{2!}x^{2}+\dfrac{f'''(0)}{3!}x^{3}+....[/tex]

f(x)=tanx

f(0)=tan0=0

[tex]f'(x)=sec^{2}x\\f'(0)=sec^{2}0=1\\f''(x)=2sec^{2}xtanx\\f''(0)=2sec^{2}0tan0=0\\f'''(x)=-4sec^{2}x+6sec^{4}x\\f'''(0)=-4sec^{2}0+6sec^{4}0=-4+6=2\\[/tex]

[tex]f''''(x)=-8(2sec^{2}xtan^{2}x+sec^{4}x)+24(4sec^{4}xtan^{2}x)+sec^{6})\\f''''(0)=-8(0+1)+24(0+1)=-8+24=16\\[/tex]

[tex]f(x)=0+x+0+\dfrac{2x^{3}}{3!}+\dfrac{16x^{4}}{4!}\\[/tex]

[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}\\[/tex]

Hence, the Taylor series for f(x)=tanx is given by

[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]

Maclaurin series upper bound error formula used is given as

R_n(x)=|f(x)-T_n(x)|

R_3(x)=|tanx-T_3(x)|

[tex]R_3(x)=|tanx-x-\dfrac{x^{3}}{3}-\dfrac{2x^{4}}{3}|[/tex]

Plugging this value x=1

[tex]R_3(x)=|tan(1)-1-\dfrac{1}{3}-\dfrac{2}{3}|\\[/tex]

R_3(x)=|1.5574-1-0.333-0.666|

R_3(x)=|-0.4426|=0.4426

Hence, upper bound on the error approximation

tan(1)=0.4426

Answer:

9.3125 hours

Step-by-step explanation:

Given:

Number of components consisting in a product = 25

Standard time for work cycle = 7.45 minutes

or

standard time for work cycle = [tex]\frac{\textup{7.45}}{\textup{60}}[/tex] hours

Number of units completed = 75

Now,

The number of standard hours

= Number of units completed × standard time for the work cycle

=  [tex]75\times\frac{\textup{7.45}}{\textup{60}}[/tex] hours

or

The number of standard hours = 9.3125 hours

Answer:

The answer is 9.3125 hours :)

Step-by-step explanation:

Step-by-step explanation:

The statement to be proved using mathematical induction is:

We will begin the proof showing that the base case is satisfied (n=4).

[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex].

Then, 1 is true for n=4.

Now we will assume that the statement holds for some arbitrary natural number [tex]n\geq 4[/tex] and prove that then, the statement holds for n+1. Observe that

[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]

With this the inductive step has been proven and then, our statement is true,

For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]

Answer: time = 55.48 years

Explanation:

Given:

Principal amount = $10000

Interest rate = 2% p.a

Amount = $30000

We can evaluate the time taken using the following formula:

[tex]Amount=Principal(1+\frac{r}{100})^{t}[/tex]

[tex]30000 = 10000\times(1+\frac{2}{100})^{t}[/tex]

Solving the above equation, we get

[tex]1.02^{t} = 3[/tex]

Now taking log on both sides, we get;

[tex]t\ log(1.02) = log(3)[/tex]

time = 55.48 years

The total volume of the bird feeder is 47.1 cubic inches.

The total volume of the bird feeder.

Given:

So, the total volume of the bird feeder is 47.1 cubic inches.

Answer: The cube root of 10 is 2.1544 using an Xo value of -0.003723

Step-by-step explanation: The Newton-Raphson is a root finding method and its formula is NR: X=Xo-(f(x)/f'(x). Once you have the equation you also need to find the derivative of that equation before applying the formula. Since the problem stated that X =10, the method was applied to find the best root in order to find the cube root of 10 up to 5 significant figures. The best method is to use a software like Excel that helps you calculate those iterations faster. The root finding for this example was -0.003723.

Answer:

[tex]Area=\frac{7}{12}[/tex]

Step-by-step explanation:

[tex]Area=\int\limits^a_b {f(x)} \, dx =\int\limits^4_3 {\frac{7}{x^{2}} } \, dx =-7*\frac{1}{x}=-7(1/4-1/3)=\frac{7}{12}[/tex]

Answer:

Area = 8

Step-by-step explanation:

A skecth is given in the attached file, there are two extra lines used to calculate the area with simple geometry:

We must use a double integral to obtain the area:

[tex]\int\limits^2_0 {\int\limits^b_a  \, dy } \, dx[/tex]

Where

b stands for y=2x+1

a stands for y=-x

Carring out the integrals we find the area:

[tex]\int\limits^2_0 {(2x+1 - (-x))} \, dx =  \int\limits^2_0 {3x+1} \, dx = (3x^{2}/2+x) \left \{ {{2} \atop {0}} \right.\\ A =( 3*2^{2}/2) + 2 =8[/tex]

Geometrically we can divide the area bounded by this lines as two triangles and a rectangle from the figure and the intersection of these lines we kno that the three figures have a base of 2. The heigth of the rectangle is 1 and for the triangles we have 4 for the upper triangle and 2 for the lower.

Therefore:

[tex]A = A_{upperT}+ A_{rect}+ A_{lowerT}[/tex]

and

[tex]A_{upperT}=2*4/2=4\\A_{rect}=2*1=2\\ A_{lowerT}=2*2/2=2[/tex]

Summing the four areas we have:

A=8

Greeting!

Step-by-step explanation:

We will prove by mathematical induction that, for every natural [tex]n\geq 4[/tex],  

[tex]5^n\geq 2^{2n+1}+100[/tex]

We will prove our base case, when n=4, to be true.

Base case:

[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex]

Inductive hypothesis:  

Given a natural [tex]n\geq 4[/tex],  

[tex]5^n\geq 2^{2n+1}+100[/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=\\=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]

With this we have proved our statement to be true for n+1.  

In conlusion, for every natural [tex]n\geq4[/tex].

[tex]5^n\geq 2^{2n+1}+100[/tex]

Answer with Step-by-step explanation:

We are given that a number 8881 is not a prime number

We have to prove that it has given a prime  factor that is at most 89.

In order to prove that given number has highest prime factor is 89 we will find the prime factorization of given number.

[tex]8881=83\times 107[/tex]

Therefore, 8881 is not  a prime number and it has two factors 83 and 107.

83 is  a prime factor of 8881 which is less than 89.We have to find  a prime factor of 8881 which is atmost 89.

Therefore, 83 is that prime factor .

Hence, 8881 has a prime factor that is at most 89.

Answer: n/2-3=13;n=32

Step-by-step explanation: the equation and solution correctly represents this sentence is n/2-3=13;n=32,

first we have to keep in mind the beginning of the exercise which is (three less) then we know that the 3 has a sign of subtraction(-3), when we meet this, we can know that this exercise has only two possible good answer.

Only the equations which have a (-3) inside will be good,the rest of equation has a 3 with a sum sign

N/2+3=13;n=5 ,

n/2+3=13;=20

then we can omit the past equations, and this let us this equations as possible.

n/2-3=13;n=32

n/2-3=13;n=8

after this we only need resolve the equation to get a correct result, only the equation with a correct result will be the correct answer,

then we proceed to clear n from each equation.

for this equation the result must be equal to 8 if we clear n.

n/2-3=13;n=8

n/2-3=13

we pass the three to the other side with sum sign

n/2=13 + 3

we resolve the sum

n/2=16

after we pass the 2 multiplying to the other side

n = 16  × 2

we resolve the product of the multiplication

n = 32

but this answer said the result must be equal to 8

n=8 ≠ n = 32

as this result is different, we can conclude that this is a bad answer,

the we get only one possibility

this equation

n/2-3=13;n=32

for this equation the result must be equal to 32 if we clear n.

n/2-3=13;n=8

n/2-3=13

we pass the three to the other side with sum sign

n/2=13 + 3

we resolve the sum

n/2=16

after we pass the 2 multiplying to the other side

n = 16  × 2

we resolve the product of the multiplication

n = 32

Answer:

C

Um I'm late but its not any of the others so....

Answer:

Ans. the amount of time needed for the sinking fund to reach $25,000 if invested $255/month at 5.8% compounded monthly (Effective monthly=0.4833%) is 80.45 months.

Step-by-step explanation:

Hi, first we need to transform that 5.8% compounded monthly into an effective monthly rate, that is as follows.

[tex]r(EffectiveMonthly)=\frac{r(Comp.Monthly)}{12} =\frac{0.058}{12} =0.00483[/tex]

That means that our effective monthly rate is 0.483%

Now, we need to solve for "n" the following formula.

[tex]FutureValue=\frac{A((1+r)^{n}-1) }{r}[/tex]

Let´s start solving

[tex]25,000=\frac{255((1.00483)^{n}-1) }{0.00483}[/tex]

[tex]\frac{25,000*0.00483}{255} =1.00483^{n} -1\\[/tex]

[tex]0.47352941=1.00483^{n} -1[/tex]

[tex]1+0.47352941=1.00483^{n[/tex]

[tex]1.47352941=1.00483^{n}[/tex]

[tex]Ln(1.47352941)=n*Ln(1.00483)[/tex]

[tex]\frac{Ln(1.47352941)}{Ln(1.00483)} =n=80.45[/tex]

This means that it will take 80.45 months to reach $25,000 with an annuity of $255 at a rate of 5.8% compounded monthly (0.4833% effective monthly).

Best of luck.

Answer:

int t(int n){

     if(n == 1)

          return 1;

     else

          return n*n*t(n-1);

}

Step-by-step explanation:

A recursive function is a function that calls itself.

I am going to give you an example of this algorithm in the C language of programming.

int t(int n){

     if(n == 1)

          return 1;

     else

          return n*n*t(n-1);

}

The function is named t. In the else clause, the function calls itself, so it is recursive.

Answer:

a) False b) True c) False d ) False e) True f) False g) False h) False

Step-by-step explanation:

a) False

x² × x³ = [tex]x^5[/tex]

When we multiply two exponential number with same base, their powers add up.

b) True

2π + 5π = π(2+5) = 7π

The coefficients of π are added together.

c) False

x² ≥ 0. For any value of x, negative or positive x² is always positive. But for x = 0, x² = 0×0 = 0

d) False

315 - 8 = 307, which is clearly an odd number.

e)True

The sum of all integers is always even.

Let m and n be two odd integers.

Thus, they can be expressed as m = 2r + 1 and n = 2s +1, where r and s are even integers.

m + n = 2r +2s + 2, which is clearly even.

f) False

Since √2 is an irrational number. It cannot belong to z, which is collection of all integer number.

√2 ∉ z

g) False

Since -1 is a negative integer, it cannot belong to [tex]z^+[/tex], as it is collection of all positive integers.

h) False

π cannot belong to Q because Q is a collection of all rational numbers and π is not a rational number. The decimal expansion of π is non- terminating that is it does not end.

Answer:

a) 0.057

b) 0.5234

c) 0.4766

Step-by-step explanation:

a)

To find the p-value if the sample average is 185, we first compute the z-score associated to this value, we use the formula

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

where

[tex]\bar x=mean\; of\;the \;sample[/tex]

[tex]\mu=mean\; established\; in\; H_0[/tex]

[tex]\sigma=standard \; deviation[/tex]

N = size of the sample.

So,

[tex]z=\frac{185-175}{20/\sqrt {10}}=1.5811[/tex]

[tex]\boxed {z=1.5811}[/tex]

As the sample suggests that the real mean could be greater than the established in the null hypothesis, then we are interested in the area under the normal curve to the right of  1.5811 and this would be your p-value.

We compute the area of the normal curve for values to the right of  1.5811 either with a table or with a computer and find that this area is equal to 0.0569 = 0.057 rounded to 3 decimals.

So the p-value is  

[tex]\boxed {p=0.057}[/tex]

b)

Since the z-score associated to an α value of 0.05 is 1.64 and the z-score of the alternative hypothesis is 1.5811 which is less than 1.64 (z critical), we cannot reject the null, so we are making a Type II error since 175 is not the true mean.

We can compute the probability of such an error following the next steps:

Step 1

Compute [tex]\bar x_{critical}[/tex]

[tex]1.64=z_{critical}=\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}[/tex]

[tex]\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}=\frac{\bar x_{critical}-175}{6.3245}=1.64\Rightarrow \bar x_{critical}=185.3721[/tex]

So we would make a Type II error if our sample mean is less than 185.3721.  

Step 2

Compute the probability that your sample mean is less than 185.3711  

[tex]P(\bar x < 185.3711)=P(z< \frac{185.3711-185}{6.3245})=P(z<0.0586)=0.5234[/tex]

So, the probability of making a Type II error is 0.5234 = 52.34%

c)

The power of a hypothesis test is 1 minus the probability of a Type II error. So, the power of the test is

1 - 0.5234 = 0.4766

The p-value is 0.014, indicating strong evidence against the null hypothesis. The probability of a type II error is 0.0907, and the power of the test is 0.9093.

To find the p-value, we need to determine the probability of observing a sample average of 185 or higher, given that the true mean foam height is 175. Since the sample size is small, we'll use the t-distribution instead of the normal distribution. With a sample size of 10, the degrees of freedom is 9. Using the t-distribution table or a calculator, we find the p-value to be 0.014 (rounded to 3 decimal places).

The probability of a type II error is the probability of failing to reject the null hypothesis when it is actually false. In this case, the null hypothesis is μ = 175, but the true mean foam height is 200. We assume α = 0.05, so the critical value is 1.645 (from the t-distribution table for a one-tailed test). Using the formula for the standard error of the sample mean, σ/√n, we can calculate the standard deviation of the sample mean to be 20/√10 = 6.32 (rounded to 2 decimal places). The difference between the critical value and the true mean is (200 - 175)/6.32 = 3.96 (rounded to 2 decimal places). Using a t-distribution table or calculator to find the area to the right of 3.96 with 9 degrees of freedom, we find the probability of a type II error to be 0.0907 (rounded to 4 decimal places).

The power of the test is 1 minus the probability of a type II error. So the power of the test is 1 - 0.0907 = 0.9093 (rounded to 4 decimal places).

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Answer: y = v₀tgθx - gx²/2v₀²cos²θ

a = v₀tgθ

b = -g/2v₀²cos²θ

Step-by-step explanation:

x = v₀ₓt

y = v₀y.t - g.t²/2

x = v₀.cosθt → t = x/v₀.cosθ

y = v₀y.t - g.t²/2

v₀y = v₀.senθ

y = v₀senθ.x/v₀cosθ - g/2.(x/v₀cosθ)²

y = v₀.tgθ.x - gx²/2v₀²cos²θ

a = v₀tgθ → constant because v₀ and θ do not change

b = - g/2v₀²cos²θ → constant because v₀, g and θ do not change

Final answer:

The trajectory of a projectile is shown to be parabolic by substituting time from the x-direction motion equation into the y-direction motion equation and rearranging, yielding a parabolic form y = ax + bx², with constants determined by initial velocity and gravity.

Explanation:

To prove that the trajectory of a projectile is parabolic, we start with the equations of motion in the x and y directions for a projectile that starts at the origin. For the x-direction, we have x = v_0x t, where v_0x is the initial velocity component in the x-direction and t is the time.

In the y-direction, the equation is y = v_0y t - (1/2) g t², where v_0y is the initial velocity component in the y-direction and g is the acceleration due to gravity.

To find t from the x equation: t = x / v_0x. Substituting this into the y equation yields: y = v_0y (x / v_0x) - (1/2) g (x / v_0x)^2.

Now, simplifying we get: y = (v_0y / v_0x) x - (g/2 v_0x^2) x^2, which is of the form y = ax + bx². Here, a = v_0y/v_0x and b = -g/2 v_0x²are constants depending on the initial velocity components and acceleration due to gravity. The equation y = ax + bx² represents a parabolic path, confirming that the projectile's trajectory is indeed a parabola.

Recall that for [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n\ge0}x^n[/tex]

Replace [tex]x[/tex] with [tex]-9x^2[/tex] and we get

[tex]\displaystyle\frac1{1-(-9x^2)}=\sum_{n\ge0}(-9x^2)^n=\sum_{n\ge0}(-9)^nx^{2n}[/tex]

Lastly, multiply this by [tex]x^3[/tex], so that

[tex]\boxed{f(x)=\displaystyle\sum_{n\ge0}(-9)^nx^{2n+3}}[/tex]

Answer:

Answered

Step-by-step explanation:

Online courses are those classes which are delivered entirely online. Students study via web cam,  chat rooms and smart boards. Whereas hybrid learning is a combination of both online learning and  traditional in class learning.  Remote work is working away from the work place at your own comfort and choice of location.