Question 4: The length of the diagonal of a square is 6 cm. Find the sides of the square.​

The maximum distance of the fire engine from the school is D = 4.5 miles

Given data ,

Distance D = √ ( x₂ - x₁ )² + ( y₂ - y₁ )²

Given data ,

Let the first point be P = P ( -4 , -3 )

Let the second point be Q = Q ( -7 , -3 )

First distance D₁ = √ ( -4 + 7 )² + ( -3 + 3 )

D₁ = √9

D₁ = 3 units

Now , each unit on the grid represents 1/2 mile

And , the distance from P to school is S ( -4 , 3 )

D₂ = √ ( -4 + 4 )² + ( -3 - 3 )²

D₂ = √36

D₂ = 6 units

So , the total distance D = 3 + 6 = 9 units

And , the distance in miles = 9/2 = 4.5 miles

Hence , the distance is 4.5 miles

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The complete question is attached below :

A fire engine takes the shortest possible route when it drives from the firehouse to one of the schools. The route can go only horizontally and vertically on the grid. Each unit on the grid represents 1-half mile. What is the maximum distance, in miles, the fire engine travels to reach one of the schools?

Answer:

  Option 3: y = 1/3x +67

Step-by-step explanation:

You want the equation of the best-fit line in slope-intercept form for the data given.

The attached calculator display shows the best-fit line for the data, where x is in minutes, is approximately ...

  y = 0.41x + 64

The closest answer choice is ...

  Option 3: y = 1/3x +67

__

Additional comment

The second attachment shows a plot of the data (study minutes, test score) with the best-fit line in red, and the Option 3 line in blue.

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The required exact values of a and b are 7 / 2 and 7√3/2 respectively. Option A is correct.

A triangle ABC is given with measures,
AB = 7, BC = a, and AC = b. Also, angle A is 30°.

To find out the missing measures, we need to apply the trigonometric ratio.

sinA = BC/AB
sin30 = a/7
1/2=a/7
a = 7 / 2

Similarly,
cosB = AC/AB
cos30 = b/7
√3/2 = b/7
b = 7√3/2

Thus, the required exact value of a and b are 7 / 2 and 7√3/2 respectively. Option A is correct.

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The number of pounds of each alloy he must use are: 49.33 of 32% aluminum and 4.67 of 44% Alloy

X = amount of alloy 1

Y = amount of alloy 2

Thus:

X + Y = 54

X = 54 - Y

0.32X + 0.44Y = 0.36 * 54

0.32X + 0.44Y = 19.44

Plugging in 54 - Y for X gives us:

0.32(59 - Y) + 0.44Y = 19.44

18.88 - 0.32Y + 0.44Y = 19.44

0.12Y = 19.44 - 18.88

Y = 0.56/0.12

Y = 4.67

X = 54 - 4.67

X = 49.33

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The measure of the unknown sides is equivalent to 12.6

The given figure is a circle geometry with the perpendicular lines intersecting both segments of the circle.

We need to determine the measure of x. Since the measure of the perpendicular lines are equal, hence the measure of the line of the segments will also be equal.

Hence:

x = 6.3 + 6.3

x = 12.6

Therefore the measure of x from the given diagram is equivalent to 12.6

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The two buses are approximately 141.2 km apart after two hours.

We have,

We can use the Law of Cosines to solve this problem.

Let's call the distance between the two buses "d".

After 2 hours, the first bus will have traveled 100 km (50 km/h x 2 h) and the second bus will have traveled 180 km (90 km/h x 2 h).

Now we can use the Law of Cosines to find d:

d² = 100² + 180² - 2(100)(180)cos(127° - 46°)

d² = 10000 + 32400 - 36000cos(81°)

d² = 42400 - 36000cos(81°)

d ≈ 141.2 km

Therefore,

The two buses are approximately 141.2 km apart after two hours.

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Answer:

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

As per definition, the cosine function is:

Since triangles are similar, the ratio of corresponding sides is same:

the volume of a prism is

length × width × height

so, for the small prism we have

5 × 4 × height = 100 in

height = 100/20 = 5 i

for the large prism we have

5 × 4 × height = 200 in³

height = 200/20 = 10 in

the larger prism is 10/5=2 times taller than the small prism.

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If 11 rain jackets were collected. At the end of the the drive, 10 times that number of rain jackets were collected. The number of rain jackets that  were sold is 99.

We must determine the total number of rain jackets gathered at the end of the campaign in order to determine how many were sold.

10 times the original quantity of raincoats gathered is:

= 10 * 11

= 110

So, 110 rain jackets were collected.

Number of raincoats sold:

= 110 - 11

= 99

Therefore 99 rain jackets were sold.

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Answer:

24.6%

Step-by-step explanation:

Use binomial probability:

P = nCr pʳ qⁿ⁻ʳ

where n is the number of trials,

r is the number of successes,

p is the probability of success,

and q is the probability of failure (1−p).

n = 9, r = 4, p = 0.5

P = ₉C₄ (0.5)⁴ (1−0.5)⁹⁻⁴

P = 126 (0.5)⁴ (0.5)⁵

P ≈ 0.246

The intensity of the Gansu earthquake was approximately 39,811 times greater than that of an earthquake measuring 4 on the Richter scale, which typically causes no damage.

Let's compare the intensities of the two earthquakes using the Richter scale.
Step 1: Understand that the Richter scale is logarithmic, which means each whole number increase represents a tenfold increase in intensity.
Step 2: Determine the difference between the magnitudes of the two earthquakes. The Gansu earthquake measured 8.6, and the earthquake causing no damage measured 4.
8.6 - 4 = 4.6
Step 3: Calculate the intensity ratio between the two earthquakes using the difference in magnitudes. Since each whole number increase represents a tenfold increase in intensity, we need to raise 10 to the power of the difference.
10^4.6 ≈ 39,810.71705535
Step 4: Round the result to a whole number.
39,810.71705535 ≈ 39,811
The intensity of the Gansu earthquake was approximately 39,811 times greater than that of an earthquake measuring 4 on the Richter scale, which typically causes no damage.

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Answer:50

Step-by-step explanation:

The number of tiles that would be grey tiles out of the total tiles given would be = 28 tiles.

To calculate the number of grey tiles, the total number of tiles given should be divided into two since the path is made up of only white and grey tiles.

The total number of tiles = 56 tiles.

Therefore the grey and the white tiles have the same number of tiles = 56/2 = 28 grey tiles

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Answer:

i think its

x^2 + (y + 2)^2 = 21.

Step-by-step explanation:

Answer:

5/2

Step-by-step explanation:

The required solution with the concern of mortgage has been shown below.
$695.67, $250,441.20, $783.93, $235179, and $15262.2.

Using the formula for the monthly payment of a mortgage:

[tex]P = L[c(1 + c)^n]/[(1 + c)^n - 1][/tex]

Here P is the monthly payment, L is the loan amount, c is the monthly interest rate, and n is the total number of payments (in this case, 30 years = 360 months).

Plugging in the values given:

L = 160000

c = 0.033/12 (monthly interest rate)

n = 360

P = 695.67

So the monthly payment is $695.67.

To find out how much will be paid in total over 30 years, we can multiply the monthly payment by the total number of payments:

Total payments = P * n = $695.67 * 360 = $250,441.20

So the total amount paid to the bank for the home over 30 years is $250,441.20.

To calculate the new monthly payment for a 25-year loan, we can use the same formula but with n = 25*12 = 300 months:

P = 160000[(0.033/12)(1 + (0.033/12))^300]/[(1 + (0.033/12))^300 - 1]

P = $783.93

So the new monthly payment for a 25-year loan is $783.93.

The total amount paid over 25 years would be:

Total payments = P * n = $783.93 * 300 = $235179

So the total amount paid to the bank for the home over 25 years is $235179.

The amount saved by reducing the time of the mortgage loan is the difference between the total amount paid for the 30-year loan and the total amount paid for the 25-year loan:

Savings = $250,441.20 - $235179 = $15262.2

So the amount saved by reducing the time of the mortgage loan is $15262.2.

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The measure of Angle YEA is 70 degrees in the given parallelogram

Given Parallelogram EASY with diagonal ES

∠AES = 40

∠Y = 110

∠ESY = 40   by  Z theorem of a parallelogram

∠A = ∠Y  by property of parallelogram

∠A = <110  as ∠Y=110

∠YES = 180 - ∠Y  

∠YES = 180 - 110 - 40  

∠YES = 30

∠YEA = ∠YES + ∠AES  

∠YEA = 30 + 40            

∠YEA = 70

Hence, the measure of Angle YEA is 70 degrees

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To represent a circle with a diameter of 12 units and a center that lies on the y-axis, we can use the standard form of the equation of a circle:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle, and r is the radius.

If the center of the circle lies on the y-axis, then the x-coordinate of the center is 0. Also, since the diameter is 12 units, the radius is 6 units.

Using these values, we can eliminate the equations that do not meet these conditions:

- x2 + (y – 3)2 = 36: This circle has a center at (0, 3), which is not on the y-axis.

- x2 + (y – 5)2 = 6: This circle has a center at (0, 5), which is not on the y-axis.

- (x – 4)² + y² = 36: This circle has a center at (4, 0), which is not on the y-axis.

- (x + 6)² + y² = 144: This circle has a center at (-6, 0), which is not on the y-axis.

- x2 + (y + 8)2 = 36: This circle has a center at (0, -8), which is on the y-axis.

Therefore, the two equations that represent circles with a diameter of 12 units and a center that lies on the y-axis are:

- x2 + (y + 8)2 = 36

- x2 + (y - 8)2 = 36

Note that the second equation is also valid, since the center of the circle can also be located at (0, -8).

The amount at which the cards offer the same deal over the course of a year is a $2,333.33

Since credit card finance charge is the fee associated with using credit.

Credit card issuers use the finance charges to minimize non-payment risks and earn some profits for extending credit to cardholders.

Data and Calculations:

                  Card A         Credit B

APR              20.8%               24.6%

Annual fee  $60                   $0

To calculate the balance required, the required equation is

= 22%x + $40 = 24.6%x

Where x = required balance

Thus,

= 0.208x + $60 = 0.246x

= $60 = 0.03x (0.246x- 0.208x )

= 0.03x = $60

x = $60 /0.03

x = $2,333.33

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The whole number for "forty-two thousand, fifteen" is 42,015.

We have, "forty-two thousand, fifteen".

To write a sentence as a whole number using digits, you would typically replace the words with their numerical representations. Here's an example:

Sentence: "I have three apples and four bananas."

Whole number: "I have 3 apples and 4 bananas."

Here the numeric terms are forty- thousand and fifteen then the numerical digit will be

= 42, 015.

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The correct statement is:

"Divide out a common factor of x - 7." The limit of the given expression is 1/6.

Multiply by (√(x + 2) + 3)/(√(x + 2) + 3):

This statement is not correct.

We generally multiply by the conjugate of the expression to eliminate square roots, but in this case, multiplying by (√(x + 2) + 3)/(√(x + 2) + 3) does not help simplify the expression.

Get x - 1 in the numerator:

This statement is not correct. The expression does not involve x - 1 in the numerator.

Get (x - 7)(√(x + 2) - 3) in the denominator:

This statement is not correct. We want to simplify the expression, not introduce a more complex denominator.

Divide out a common factor of x - 7:

This statement is correct. By factoring out (x - 7) from both the numerator and denominator, we can cancel out the common factor.

Calculate the limit as 1/6:

This statement is not correct. We need to perform further simplification to determine the limit.

After canceling out the common factor of x - 7, the simplified expression becomes:

lim x -> 7 (√(x + 2) - 3)/(x - 7) = lim x -> 7 1/(√(x + 2) + 3)

To find the limit, we substitute the value x = 7 into the simplified expression:

lim x -> 7 1/(√(7 + 2) + 3) = 1/(√(9) + 3) = 1/(3 + 3) = 1/6

Therefore, the correct statement is: "Divide out a common factor of x - 7." The limit of the given expression is 1/6.

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if y varies inversely as the same of x, and y=7/4 when x = 1 then value of y is 7/12 when x=3

If y varies inversely as the same of x, we can write the equation:

y = k / x

where k is the constant of proportionality.

Let us find the value of constant of proportionality k:

y = 7/4 when x = 1

7/4 = k / 1

Apply cross multiplication we get:

7/4 = k

k=7/4

Now plug in the value of k to find y when x = 3:

y = k /x

y = (7/4) / 3

When a fraction is divided by the whole number then fraction is multiplied with reciprocal of the whole number.

y=7/4×1/3

Multiply the numerators and denominators.

y=7×1/4×3

We know that 7×1  is 7 and 4×3 is 12.

y=7/12

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ŷ= 6.45x + 31.5 is the equation of the regression line for the given data

Here, we have,

to find the equation of the regression line for a given data:

Construct the table of the data as follows:

x    >>>>     y  >>>>  x²  >>>>  y²  >>>>   xy

1                39            1            1521         39

2               45            4           2025       90

3               52             9          2704       156

4               49            16          2401        196

5               61             25         4096       305

5               72            25          5184        360

................................................................................................

20             318           80          17931       1146

.................................................................................................

∑x = 20,  ∑y = 318, ∑x²= 80, ∑xy = 1146,  n = 6 (number data points)

The linear regression equation is of the form:

ŷ = ax + b

where a and b are the slope and y-intercept respectively

a = ( n∑xy -(∑x)(∑y) ) / ( n∑x² - (∑x)² )

a = (6×1146 - 20×318) / ( 6×80-20² )

a = 6876-6360 / 480-400

a = 516 / 80

a = 6.45

x' =  ∑x/n

x' = 20/6 = 10/3

y' = ∑y/n

y' = 318/6 = 53

b = y' - ax'

b = 53 - 6.45×(10/3)

b = 53 - 21.5

b = 31.5

ŷ = ax + b

ŷ= 6.45x + 31.5

Thus,  the equation of the regression line for the given data is

ŷ= 6.45x + 31.5.

(a) x=3 hours

ŷ= 6.45(3) + 31.5 = 50.85

(b) x=4.5 hours

ŷ= 6.45(4.5) + 31.5 = 60.525

(c) x=13 hours

ŷ= 6.45(13) + 31.5 = 115.35

(d) x=1.5 hours

ŷ= 6.45(1.5) + 31.5 = 41.175

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complete question:

Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line.(The pair of variables have a significant correlation.) Then use the regression equation to predict the value of y for each of the given x-values, if meaningful. The number of hours 6 students spent for a test and their scores on that test are shown below.

(a) x=3 hours

(b) x=4.5 hours

(c) x=13 hours

(d) x=1.5 hours

The total number of beads in the two boxes: is: 20

The parameters given are:

Number of white beads in box A = 7

Number of black beads in box A = x - 7

Number of white beads in box B = 4

Number of black beads in box B = x - 4

The probability of getting white from both boxes is:

P(W & W) = (7/x) * (4 + 1)/(x + 1)

P(W & W) = 35/(x(x + 1)

We are given that P(W & W) = 7/22

Thus:

35/(x(x + 1) = 7/22

Solving gives x = 10

Thus:

Box A has 10 beads and box B has 10 beads.

Total number of beads = 20 beads

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Answer:

Sure, here are the steps on how to simplify 2/3 (3x -2) – ¾ (2x -2):

Multiply the factors in the numerator and denominator of each fraction.

2/3 (3x -2) = (2)(3x -2) / (3)(3) = 6x - 4 / 9

¾ (2x -2) = (3/4)(2x -2) / (3) = 3x - 6 / 12

Subtract the numerators of the two expressions.

6x - 4 / 9 - 3x - 6 / 12 = (6x - 4 - 3x - 6) / (9 * 12) = 3x - 10 / 108

Simplify the fraction by dividing the numerator and denominator by 3.

3x - 10 / 108 = (3x - 10) / (3 * 36) = x - 10 / 36

Therefore, the simplified expression is x - 10 / 36.

Step-by-step explanation:

Answer:

slope = [tex]\frac{5}{9}[/tex] , y- intercept = - 5

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

5x - 9y = 45 ( subtract 5x from both sides )

- 9y = - 5x + 45 ( divide through by - 9 )

y = [tex]\frac{-5}{-9}[/tex] x + [tex]\frac{45}{-9}[/tex] , that is

y = [tex]\frac{5}{9}[/tex] x - 5 ← in slope- intercept form

with slope m = [tex]\frac{5}{9}[/tex] and y- intercept c = - 5