How many possible passwords can be made for a website that requires passwords to consist of 3 digits followed by 2 letters? (Note that each digit or letter is allowed to be repeated.

Answers

Answer 1

(I would appreciate a Brainliest rating if this helped you)

(I would appreciate a Brainliest rating if this helped you)Answer:

456,976,000 possible passwords

Step-by-step explanation:

To calculate the number of possible passwords, we need to figure out how many options we have for each character slot in the password.

For the first three slots, we need to choose a digit from 0 to 9. This means we have 10 options for each slot, since there are 10 digits to choose from. Since there are three slots for digits, the number of possible combinations for the first three characters is 10 x 10 x 10 = 1000.

For the next four slots, we need to choose a letter from A to Z. There are 26 letters in the alphabet, so we have 26 options for each slot. Since there are four slots for letters, the number of possible combinations for the last four characters is 26 x 26 x 26 x 26 = 456,976.

To find the total number of possible passwords, we simply multiply the number of combinations for the first three characters by the number of combinations for the last four characters:

1000 x 456,976 = 456,976,000

Therefore, there are 456,976,000 possible passwords that can be made for this website.

Related Questions

The probability that a tennis set will go to a tiebreaker is â€‹14%. What is the probability that two of three sets will go toâ€‹ tie-breakers? Round the answer to the nearest thousandth.

Answers

The probability that two out of three sets will go to tiebreakers ≈ 0.061

To calculate the probability that two out of three sets will go to tiebreakers, we need to consider the possible combinations of sets that can have tiebreakers.

There are three possible scenarios where two sets out of three can have tiebreakers:

1. The first two sets go to tiebreakers, and the third set does not.

2. The first and third sets go to tiebreakers, and the second set does not.

3. The second and third sets go to tiebreakers, and the first set does not.

The probability of each scenario occurring can be calculated by multiplying the probability of a tiebreaker (0.14) by the probability of not having a tiebreaker (1 - 0.14 = 0.86) for the remaining set.

1. Probability of scenario 1: (0.14) * (0.14) * (0.86) = 0.020408

2. Probability of scenario 2: (0.14) * (0.86) * (0.14) = 0.020408

3. Probability of scenario 3: (0.86) * (0.14) * (0.14) = 0.020408

To find the probability that any of these scenarios occur, we sum up the probabilities of the three scenarios:

Total probability = 0.020408 + 0.020408 + 0.020408 = 0.061224

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each point of the plane is colored red or blue. show that there is a rectangle whose corners are all the same color

Answers

We can always find a monochromatic rectangle by reducing the problem to finding a monochromatic rectangle in a smaller area by moving the bottom of the rectangle up by one unit method.

What is rectangle?

A rectangle is a quadrilateral (a 2-dimensional shape with four sides) with four right angles. This means that opposite sides of a rectangle are parallel and congruent, and all four angles are equal to 90 degrees.

Let us consider a rectangle with sides parallel to the coordinate axes. Such a rectangle can be uniquely determined by two pairs of points that define its opposite corners. If all four of these points are the same color, then we have found a monochromatic rectangle. Otherwise, there are three cases to consider:

The two points at the top of the rectangle are the same color, and the two points at the bottom are the other color. In this case, we can reduce the problem to finding a monochromatic rectangle in a smaller area by moving the bottom of the rectangle up by one unit.

The two points on the left side of the rectangle are the same color, and the two points on the right side are the other color. In this case, we can reduce the problem to finding a monochromatic rectangle in a smaller area by moving the left side of the rectangle to the right by one unit.

Both pairs of points are of mixed color. In this case, we can reduce the problem to finding a monochromatic rectangle in two smaller areas by dividing the rectangle into four equal parts with a vertical or horizontal line.

By repeating this process on the smaller rectangles obtained in each case, we can eventually find a monochromatic rectangle. Since the rectangles we consider at each step have half the area of the previous ones, this process terminates after at most log_2(A) steps, where A is the area of the original rectangle.

Therefore, we can always find a monochromatic rectangle with this method.

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Complete question:

Each point in the x-y plane colored red or blue. show that there is a rectangle whose corners are all the same color.

find an equation of the line that is perpendicular to the line y=5x+6 and passes through the point (-2,5)
workings pls

Answers

Answer:

$5y=-x+23\\$

Step-by-step explanation:

Since the equation of the line is perpendicular to the line y = 5x +6

we can use the formula:

$m_1m_2=-1\\$

where m1 is the slope of the line we have and,

m2 is the slope of the line we need

so here m1 is 5 as we compare it with the slope-intercept form our slope is:

$y=5x+6\\y=mx+b$

our m1 is 5 but we need m2 so plugging in the value we get m2 to be:

$m_1m_2=-1\\5m_2=-1\\m_2=-1/5\\$

and now we have our m2, now furthermore since the line passes through the point (-2 , 5) it means the point must lie on the line and satisfy it. We can use this point and the slope m2 and find out the equation by the point-slope form equation:

$y-y_1=m(x-x_1)\\y-5=-\frac{1}{5}(x-(-2))\\\\5(y-5)=(-1)(x+2)\\5y-25=-x-2\\5y=-x-2+25\\5y=-x+23\\$

8=8p+13−3p please help

Answers

Answer:

-1=p

Step-by-step explanation:

8=8p+13-3p

8-13=8p-3p

-5=5p

-5÷5=5p÷5

-1=p

jennas dogs used 9 chew toys last year. The animal shelter used 2,277 chew toys. How many times as many chew toys did the animal shelter use?

Answers

253

2,277 divided by 9=253

hope this helps

Answer:

253 is the answer

Step-by-step explanation:

Divide 2,277 and 9.

Dr. Aghedo is saving money in an account with continuously compounded interest. How long
will it take for the money she deposited to double if interest is compounded continuously at a
rate of 3.1%. Round your answer to the nearest tenth.
O a
C
Od
17.8 years
12.5 years
22.4 years
10.2 years

Answers

The time required for the deposit to double at 3.1% interest rate is 22.4 years.

What is the time required for the deposit to double?

The formula for calculating compound interest can expressed as;

A = Pe^( rt )

Where A is the accrued amount, P is principal, r is interest and t is time.

Given the data in the question;

Let the deposit be represented by x

Principal P = x

Since the deposit doubled

Accrued amount A = 2xCompounded continuouslyInterest rate r = 3.1%Time t = ?

Plug the values into the above compound interest formula and solve for time t.

A = Pe^( rt )

t = ( In( A / p ) ) / r

t = ( In( 2x / x ) ) / 3.1%

t = ( In( 2 ) ) / 0.031

t = 22.3595

t = 22.4 years.

Therefore, the time needed is 22.4 years.

Option C is the correct answer.

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plsss help!!!!!!!!

Answers

Answer:

Step-by-step explanation:

so the area of a circle is A=$\pi r^{2}$

111.6=3.14*$r^{2}$ divide by 3.14 on both sides. that makes 36=$r^{2}$ take the square root of both sides. r=6.

now we find the diameter. 2\pi r=2*3.14*6=3.14*12 which is approximately 38.68

In a study of the performance of a new engine design, the weight of 22 aircrafts (in tons) and the top speed (in mph) were recorded. A regression line was generated and shown to be an appropriate description of the relationship. The results of the regression analysis are below. Depend Variable: Top Speed Variable Constant Weight Coefficient 11.6559 3.47812 s.e. of Coeff 0.3153 0.294 t-ratio 37 11.8 prob ≤ 0.0001 ≤ 0.0001 R squared = 87.5% R squared (adjusted) = 86.9% s = 0.6174 with 22 - 2 = 20 degrees of freedom Part A: Provide the regression equation based off the analysis provided and explain it in context. (2 points) Part B: List the conditions for inference that need to be verified. Assuming these conditions have been met, does the data provide convincing evidence of a relationship between weight and top speed? (4 points) Part C: Assuming all conditions for inference have been verified, determine a 95% confidence interval estimate for the slope of the regression line. (4 points)

Answers

The predicted value for the top speed of an aircraft if its weight is 100 tons = 359.4679 mph.

There is convincing evidence of a relationship between weight and top speed, assuming the conditions for inference have been met

We are 95% confident that the true slope of the regression line is between 2.871 and 4.085.

How to solve

Part A)

Independent variable: X: Weight

Response variable: Y: Top speed of an aircraft.

Slope = b = 3.47812

Y-intercept = a = 11.6559

LSRL based on analysis is,

$\hat{Top\ speed}=11.6559+3.47812\ Weight$

Part B):

Here, p≤ 0.0001 indicates that there linear relationship between two variables. Therefore, we can use LSRL to predict top speed.

Given: Weight = x = 100 tons.

Therefore,

$\hat{Top\ speed}=11.6559+3.47812*100$

$\hat{Top\ speed}=359.4679\ mph$

Hence, the predicted value for the top speed of an aircraft if its weight is 100 tons = 359.4679 mph.

Part B:

Conditions for inference:

Linearity: The relationship between weight and top speed is linear.

Independence: The aircrafts' weights and top speeds are independent observations.

Normality: The residuals have a normal distribution.

Equal variance: The residuals have constant variance.

Part C:

To calculate the 95% confidence interval for the slope, we use the formula: slope ± t_critical * s.e. of Coeff.

With 20 degrees of freedom, the t_critical value is approximately 2.086. So, the 95% CI for the slope is: 3.47812 ± (2.086 * 0.294) = (2.871, 4.085).

This means we are 95% confident that the true slope of the regression line is between 2.871 and 4.085.

Answers

Answer:

3 centimeters

Step-by-step explanation:

You can find this by dividing 54 by 18, or if you know your timetables, you know that 18x3 is 54.

Answer:

Step-by-step explanation:

Just compare like things and write a proportion is one way to think of it. It helps to recognize that 18×3=54, but you might not know that automatically bc it's not on the times table from elementary school.

A map represents every 18 kilometers with 1 centimeter. If two locations are actually 54 kilometers apart,

What is the equation of the given line in standard form? use integer coefficients. y=-1.7x 8.5

Answers

The given equation, y = -1.7x + 8.5, is not in standard form with integer coefficients. The standard form of a linear equation is Ax + By = C, where A, B, and C are integers and A is positive.

To convert the given equation to standard form, we need to eliminate the decimal coefficient. We can do this by multiplying both sides of the equation by 10 to clear the decimal: 10y = -17x + 85 Next, we want the coefficient of x to be positive, so we can multiply both sides of the equation by -1: -10y = 17x - 85

Now, we can rearrange the terms to match the standard form: 17x - 10y = 85 So, the equation of the given line in standard form with integer coefficients is 17x - 10y = 85. To convert the given equation to standard form, we need to eliminate the decimal coefficient.

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What is the solution to the equation below? Round your answer to twodecimal places.eX = 7.9

Answers

Solution

For this case we have the following equation:

$e^x=7.9$

For this case we can apply natural log in both sides and we got:

$\ln (x)=\ln (7.9)=2.07$

and then the solution would be:

D. x =2.07

What is the mean absolute deviation for Doctor A’s data set on glasses? What is the mean absolute deviation for Doctor B’s data set on glasses? Write a sentence comparing the variation of the two data sets using their mean absolute deviations.

Answers

There is more variability on glasses of doctor B's dataset than the glasses of doctor B's dataset

Doctor A’s mean absolute deviation on glasses

The dataset for this is given as:

643 634 670 658 636 624 641 655 649 629

Using a statistical calculator, we have:

Size = 10Mean (μ) = 643.9Mean Absolute Deviation (MAD) = 11.28

Hence, the mean absolute deviation for Doctor A’s data set on glasses is 11.28

Doctor B’s mean absolute deviation on glasses

The dataset for this is given as:

651 625 622 624 631 621 653 647 646 646

Using a statistical calculator, we have:

Size = 10Mean (μ) = 636.6Mean Absolute Deviation (MAD) = 12

Hence, the mean absolute deviation for Doctor B’s data set on glasses is 12

The variation of the two data sets using their mean absolute deviations.

In a dataset, the larger the mean absolute deviations value, the larger the variation.

By comparison, 12 is greater than 11.28

Hence, there is more variability on glasses of doctor B's dataset than the glasses of doctor B's dataset

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There were 32 students in Jaden’s class eating lunch. Then, more students joined Jaden’s class. Now there are 86 total students eating lunch. How many students joined Jaden’s class?

Answers

Answer

Step-by-step explanation:

86 - 32 = 54

so 54 students joined Jadens class

Answer:

54 students joined Jaden's class

Step-by-step explanation:

86 - 32 = 54

In the equation, it states that there are 32 students in Jaden's class at first. If more join, and now there are 86 students, you need to subtract 32 from 86.

2r² +23r+56=0
Solve by factoring

Answers

mid-term factorization

Use the quadratic eqation:

-B +/- $\sqrt{B^{2}-4AC }$/2A

-23 +/- $\sqrt{23^{2} -(4 )(2)(56) }$/ (2) (2)

-23 +/- 9/4

Two roots are:

-23 + 9/4 = -7/2

and

-23- 9/4 = -8

Akira ordered a set of beads. She received 100 beads in all. 8 of the beads were green. What percentage of the beads were green?

Answers

Answer:

Step-by-step explanation:

8 out of 100 are green

8/100 * 100% = 8 %

Answer:

8% of the beads are green

Step-by-step explanation:

Step 1: Determine what percentage of the beads are green

This problem is known as a probability problem which basically tries to determine the proportionality between certain numbers. In this example we are trying to figure out the proportionality between the green beads and the total sample.

(8 green beads) / (100 total beads)

0.08

0.08 is the probability but if we want to determine what the percentage is of the number we just multiply it by 100.

0.08 * 100 = 8%

Answer: 8% of the beads are green

Find the complete solution of the linear system, or show that it is inconsistent. (If the system has infinitely many solutions, express your answer in terms of t, where x = x(t), y = y(t), and z = t. If there is no solution, enter NO SOLUTION.) x ? 2y + z = 3 2x ? 5y + 6z = 7 2x ? 3y ? 2z = 5 (x, y, z) =

Answers

The solution of a linear equation is (-2,5,4).

The equations given in the questions are,

x - y + 2z = 1 ..[1]

3x + y + 5z = 19 ..[2]

2x - y - 2z = -17 ...[3]

Putting the value of y from [1] in [2]and [3]

y = x+ 2z -1

3x + (x+ 2z -1) + 5z = 19

4x + 7z = 20 ...[4]

2x - (x+ 2z -1) - 2z = -17

x - 4z = -18 ...[5]

Solving equations [4] and [5] by substitution:

Putting values of x from [5] in [4]

x = 4z -14

4(4z -18) + 7z = 20

16z - 72+ 7z =20

23z = 92

z = 4

x = 4 × (92 ÷ 23) - 18

x = -2

y = (-2) + 2 × (92 ÷ 23) - 1

y = 5

(x,y,z) = (5.-2,4)

The solution of the equation is (-2,5,4).

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The function h is defined by
h(x)=x²-2.
Find h(2z)

Answers

Answer:

$h(x) = {x}^{2} - 2 \\ h(2z) = {(2z)}^{2} - 2 \\ = 4 {z}^{2} - 2 \\ = 2( {2z}^{2} - 1)$

Please help for section d) 100 points, must show all working and step by step

Answers

Answer:

Step-by-step explanation:

(a) and (b) see diagram

(d) Solving simultaneously can mean to set equal

6x - x² = k >subtract k from both sides

6x - x² - k = 0 >put in standard form

- x² + 6x - k = 0 >divide both sides by a -1

x² - 6x + k = 0

(e) The new equation is the same as the original equation just flipped (see image)

(f) The discriminant is the part of the quadratic equation that is under the root. (not sure if they wanted the discriminant of new equation or orginal. I chose new)

discriminant formula = b² - 4ac

equation: x² - 6x + 6 = 0 a = 1 b=-6 c = 6

discriminant = b² - 4ac

discriminant= (-6)² - 4(1)(6)

discriminant = 36-24

discriminant = 12

Because the discriminant is positive, if you put it back in to the quadratic equation, you will get 2 real solutions.

Based on the diagram below, what is the measure of arc DC?

28 degrees
68 degrees
108 degrees
112 degrees

Answers

Answer:

108 °

Step-by-step explanation:

∠DOC = 180° - 112° = 68°

∠DOC = 1/2(arc AB + arc DC)

68 = 1/2(y + 80 + y)

68 = y + 40

y = 28

arc dc = 80 + 28 = 108 °

The library is 5 miles from the post office. How many yards is the library from the post office?

a 5 yards
b 352 yards
c 8,800 yards
d 17,600 yards

Answers

Answer:

c 8,800 yards

Step-by-step explanation:

1 mile = 1760 yards

5 miles = 5 * 1760 yards = 8,800 yards

Answer: c 8,800 yards

answer is c 8,800 yards

What is the least common denominator of 5/12 and 1/5

Answers

Answer:

the lcd for both is 60 5/12 is 25/60 and 1/5 is 12/60

Step-by-step explanation:

Hello and Good Morning/Afternoon:

Let's take this problem step-by-step:

What does this problem want us to solve:

$\hookrightarrow \text{Find least common denominator of 5/12 and 1/5}$

Essentially:

$\hookrightarrow \text{Find the least common multiple or LCM of 5 and 12}$

The LCM of 5 and 12:

$\hookrightarrow \text{is 60}$

Answer: 60 is the least common denominator of 5 and 12

$\frac{5}{12}=\frac{5}{12}*1=\frac{5}{12}*\frac{5}{5} =\frac{25}{60}$$\frac{1}{5}=\frac{1}{5}*1 = \frac{1}{5}*\frac{12}{12} =\frac{12}{60}$

Terms used:

Least common multiple: the smallest multiple shared by the two numbers

Hope that helps!

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before. (Up to a sign that depends on whether the atoms are ous e'll discuss this aspect in more detail in Sections 3.5 and 3.6 ). How an this into account - we wrote the expression Z=Z1N which wo N particles in the were distinguishable - for example, if each of t different type. But this naive partition function overcounts the he system when we're dealing with indistinguishable particles. is a simple matter to write down the partition function for N icles. We simply need to divide by the number of ways to perm ther words, for the ideal gas the partition function is Zideal (N,V,T)=N!1Z1N=N!λ3NVN extra factor of N ! doesn't change the calculations of pressure o , we had to differentiate logZ and any overall factor drops out. age the entropy since this is given by, ∑S=∂T∂(kBTlogZideal )∂t∂(llc h includes a factor of logZ without any derivative. Of course, unting the number of underlying microstates, we would expect ther particles are distinguishable or indistinguishable. Using th tion (2.12) and Stirling's formula, the entropy of an ideal gas is S=NκB[log(Nλ3V)+25] result is known as the Sackur-Tetrode equation. Notice th py sensitive to the indistinguishability of the particles, but owever, the entropy is not directly measurable classically. py differences by the integrating the heat capacity as in (1.10

Answers

The partition function of an ideal gas can be calculated using the following expression Zideal (N,V,T)=N!1Z1N=N!λ3NVN. The extra factor of N! doesn't change the calculations of pressure or volume because we had to differentiate logZ and any overall factor drops out. We can also calculate the entropy of an ideal gas using the following S=NκB[log(Nλ3V)+25]. This result is known as the Sackur-Tetrode equation.

Notice that the entropy is not directly measurable classically, but we can measure entropy differences by integrating the heat capacity as in (1.10).When dealing with distinguishable particles, we can write the expression Z=Z1N which would work for N particles that were distinguishable.

However, this naive partition function overcounts the system when we're dealing with indistinguishable particles. It is a simple matter to write down the partition function for N identical particles. We simply need to divide by the number of ways to permute them. In other words, for the ideal gas.

the partition function is Zideal(N,V,T)=N!1Z1N=N!λ3NVN. The entropy of an ideal gas can be calculated using the formula S=NκB[log(Nλ3V)+25]. Note that this result is known as the Sackur-Tetrode equation.

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Two cars travel at the same speed to different destinations. Car A reaches its destination in 12 minutes. Car B reaches its destination in 18 minutes. Car B travels 4 miles farther than Car A. How fast do the cars travel?

Answers

Answer:
speed of car A =Speed of car B=0.8 miles/minutes.
Step-by-step explanation:
We are given that speed of car A is equal to speed of car B.
Also let car A travels x miles.
and car B travels y miles.
Car A reaches its destination in 17 minutes.
this means that speed of car A is given by: miles/minutes ( since speed is defined as the ratio of distance and time).
Car B reaches its destination in 32 minutes.
This means that the speed of car B is given by: miles/minutes
as speed of both cars are equal this means:
------(1)
Also we are given Car B travels 12 miles farther than Car A.
this means
------(2)
on using equation (1) and (2) we have:

Hence the speed of car A is 0.8 miles/minutes ( since x/17 is the speed of car A)
Speed of car B=0.8 miles/minutes.

Consider Y= F(x) =2- 3x. using the space determine the equation that represents the inverse of f(x)

Answers

Answer: $f^{-1}(\text{x}) = \frac{2-\text{x}}{3}\\\\$

===============================================

Work Shown:

$f(\text{x}) = 2-3\text{x}\\\\\text{y} = 2 - 3\text{x}\\\\\text{x} = 2 - 3\text{y}\\\\\text{x} + 3\text{y} = 2\\\\3\text{y} = 2-\text{x}\\\\\text{y} = \frac{2-\text{x}}{3}\\\\f^{-1}(\text{x}) = \frac{2-\text{x}}{3}\\\\$

-----------------

Explanation:

The process to find the inverse involves this outline.

Replace f(x) with y.Swap x and y.Solve for y.

This process is followed in the "work shown" portion above.

To confirm the answer, let g(x) be the inverse of f(x).

If they were inverses of each other, then these two properties would hold true:

f( g(x) ) = xg( f(x) ) = x

Here's the verification for the first part.

$f(\text{x}) = 2-3\text{x}\\\\f(g(\text{x})) = 2-3*g(\text{x})\\\\f(g(\text{x})) = 2-3*\frac{2-\text{x}}{3}\\\\f(g(\text{x})) = 2-(2-\text{x})\\\\f(g(\text{x})) = 2-2+\text{x}\\\\f(g(\text{x})) = \text{x}\\\\$

I'll let you do the other confirmation.

A line passes through the points (4,19) and (8,23) Write a linear function rule in terms of x and y for this line.

the linear function rule = y __

Answers

The equation of line passing through points $(x_{1} ,y_{1} )$ = (4,19) and $(x_{2},y_{2})$ =

( 8,23 ) is represented in the slope intercepted form as y = x + 15

What is the Slope Intercept Form ?

When you know the slope of the line to be investigated and the given point is also the y intercept, you can utilise the slope intercept formula,

y = mx + b. (0, b) The symbol b in the formula stands for the y value of the y intercept point.

According to the given information

We are given with two points

$(x_{1} ,y_{1} )$ = (4,19)

$(x_{2},y_{2})$ = ( 8,23 )

Now we need to find the slope of the given line having the above coordinate points

m = $\frac{y_{2} -y_{1} }{x_{2}-x_{1} }$

Putting the values

We get

m = $\frac{23-19}{8-4}$

m = $\frac{4}{4}$

m = 1

Equation of line passing through points $(x_{1} ,y_{1} )$ = (4,19) and having

slope = 1

$y-y_{1} = m(x-x_{1})$

y - 19 = x - 4

y = x + 15

The equation of line passing through points $(x_{1} ,y_{1} )$ = (4,19) and $(x_{2},y_{2})$ =

( 8,23 ) is represented in the slope intercepted form as y = x + 15

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The diagram beloon shows a cup in the shape of a right circular conse and yone of its measurements,
cm
10 cm
Which of the following is closest to the volume, in cubic centimeters, of the cup?

Answers

Answer:

option c

volume of right circular cone=1/3 *(pi)* r*r *h

Which is a true statement about the transformation?

A.) Triangle ABC was translated 8 units left and 3 units up.

B.) The pre-image is in quadrant II.

C.) The orientation of the figure was changed.

D.) The orientation of the vertices was changed
HELP ASAP

Answers

Answer: the answer is B

Step-by-step explanation:

Answer:b

Step-by-step explanation:

As seen in the accompanying diagram, a person can travel from New York City to Bufalo by going north 170 miles to Albany and then west 280 miles to Buffalo Buffalo If an engineer wants to design way to connect New York City directly to Buffalo what angle, would she need to build the highway? Find the angle to the nearest degree. To the serie how many miles would be saved by traveling directly from New York City to Buffalo rather than by traveling first to Albany and then to Buffalo

Answers

Anglex is 58.74 degre.

Miles saved by travelling in highways is 122.43 miles.

Given:

Distance between New York to Alban is 170 miles.

Distance between Alban to Baffalo is 280 miles.

The objective is to find the angle of highway to be built directly from New York to Buffalo. and also the difference in mile between the two routes.

The given image can be represented as,

In the obtained triangle right angled at B, as angle is required at A, then AC represents the hypotenuse side, AB represents the adjacent side and BC represents the opposite side.

Now,the angle can be calculated by the opposite side and adjacent side using tan theta.

$\begin{gathered} \tan \theta=\frac{opposite}{adjacent} \\ \tan x=\frac{280}{170} \\ x=\tan ^{-1}(\frac{280}{170}) \\ x=59^0 \end{gathered}$

Now, the difference between the two routes can be calculated using Pythagorean theorem of the right triangle.

$\begin{gathered} \text{Hypotenuse}^2=Opposite^2+Adjacent^2 \\ AC^2=BC^2+AC^2 \\ AC^2=280^2+170^2 \\ AC^2=78400+28900 \\ AC^2=107300 \\ AC=\sqrt[]{107300} \\ AC=327.57\text{ miles} \end{gathered}$

Thus the distance of highways directly from New york to Buffalo is 327.57 miles.

Now, consider D as the difference miles that would be saved by traveling directly from New York City to Buffalo

Then, the miles saced by travelling in this highway is,

$\begin{gathered} D=(AB+BC)-AC \\ D=(280+170)-327.57 \\ D=450-327.57 \\ D=122.43\text{ miles} \end{gathered}$

Hence, the angle required to build the highway is 59 degree. And by traveling in this highway directly from New york city to Buffalo will save 122.43 miles.