What is the length of each leg of a 45-45-90 triangle if the hypotenuse has a length of 4√2?

On solving the provided question, we can say that equation, k(x) = 4(x + 2) represents the function where you add 2 to the input, then multiply the result by 4.

A mathematical equation is a formula that links two assertions and signifies equivalence with the equals sign (=). In algebra, a mathematical statement that proves the equality of two mathematical expressions is referred to as an equation. The equal sign, for instance, separates the variables 3x + 5 and 14 in the equation 3x + 5 = 14. There is a mathematical formula that explains the link between the two phrases that appear on either side of a letter. Frequently, the symbol and the single variable are the same. like in 2x - 4 = 2, for example.

a. f(x) = 2x+4 represents the function where you multiply the input by 2, then add 4 to the result.

b. g(x) = 2(x+4) represents the function where you add 4 to the input, then multiply the result by 2.

c. h(x) = 4x+2 represents the function where you multiply the input by 4, then add 2 to the result.

d. k(x) = 4(x + 2) represents the function where you add 2 to the input, then multiply the result by 4.

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The people having to be in a room in order that the probability that at least two of them celebrate their birthday in the same month is at least 1/2 is 5.

X = No of people having same month birth day

We need, P(X 2) ≥ 1/2

= 1-P(X ≤1) ≥ 1/2

1-P(No one having same month birth day) ≥ 1/2

P(No one having same month birth day)≤ 1/2..........................(1)

Now, k = Minimum no of people required to satisfy (1)

P(No one having same month birth day) =

[tex]\frac{11*10*9......*(12-k+1)}{12*12*12......*12}[/tex]

for k= 4

P(No one having same month birth day) =

[tex]\frac{11.10.9}{12.12.12} = 0.572 \geq 0.5[/tex]

for k= 5

P(No one having same month birth day) =

[tex]\frac{11.10.9.8}{12.12.12.12} = 0.38 \leq 0.5[/tex]

Therefore , the people having to be in a room in order that the probability that at least two of them celebrate their birthday in the same month is at least 1/2 is 5.

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The centroid of the given triangle is [tex]( 2\frac{2}{3} , -3\frac{1}{3} )[/tex], which is option B.

What is a centroid of a triangle?

A place where a triangle's medians coincide is the centroid of the triangle. The line segments that are drawn from a vertex to the midpoint of the vertex's opposite side are known as medians.

A triangle is divided into two smaller triangles with equal areas by each of its medians. The centroid of a triangle is where the medians of the triangle cross. Unlike other points of concurrency in a triangle, the centroid always lies inside one.

The formula of the centroid of a triangle when the coordinates of the vertices are known is:

Centroid = [tex](\frac{x_1 +x_2+x_3}{3} , \frac{y_1 +y_2+y_3}{3})[/tex]

From the graph, we can find the vertices of the triangle

A = (x₁, y₁) = (-3,-7)

B = (x₂,y₂) = (5,1)

C = (x₃,y₃) = (6,-4)

Centroid = [tex](\frac{x_1 +x_2+x_3}{3} , \frac{y_1 +y_2+y_3}{3}) = (\frac{-3+5+6}{3} ,\frac{-7+1-4}{3}) = (\frac{8}{3} ,\frac{-10}{3} ) = ( 2\frac{2}{3} , -3\frac{1}{3} )[/tex]

Therefore the centroid of the given triangle is [tex]( 2\frac{2}{3} , -3\frac{1}{3} )[/tex].

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[tex]above \: the \: line[/tex]

The amount of money that is take home pay would be = $3,491.91

The amount of money that is your fix expenses would be = $1,257.09

The amount of money that you make each month = $3,764.82.

The percentage of deduction made for FICA = 7.25%

That is;

= 7.25/100 × 3,764.82/1

= 27294.945/100

= $272.91

Therefore, the take home pay ;

= $3,764.82-$272.91

= $3,491.91

The percentage of the fixed expenses = 36% of the realised income.

That is, 36/100 × 3,491.91/1

= 125708.76/100

= $1,257.09.

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The height of the plane is 1023 meters and the height of the tree is 307 feet.

Given the data in question 9, this forms a right-triangle.

To determine the height of the plane x, we use trigonometric ratio

Tangent = Opposite / Adjacent

tan( 12 ) = x / 4812m

x = tan( 12 ) × 4812m

x = 1023m

Therefore, the height of the plane is 1023 meters.

Option A) 1023 meters is the correct answer.

Given the data in question (10)

Solve for x using trigonometric ratio,

Tangent = Opposite / Adjacent

tan( 62 ) = x / 160ft

x = tan( 62 ) × 160ft

x = 301ft

Now, height of the tree will be;

Height = x + height of tool

Height = 301ft + 6ft

Height = 307 ft

Therefore, the height of the tree is 307 feet.

Option D) 307 feet is the correct answer.

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The function that has a simplified base of (4∛4) is 4(∛16[tex])^({2x)[/tex].

Calculating the exponential growth or decay of a given collection of data is done using an exponential function, which is a mathematical function. Exponential functions, for instance, can be used to estimate population changes, loan interest rates, bacterial growth, radioactive decay, and disease spread.

From the general formula of an exponential function, we know that;

f(x) = a[tex](b)^x[/tex]

Now, we have;

f(x) = (4∛4[tex])^x[/tex]

4(∛16[tex])^({2x)[/tex], the base is (∛16)²

(∛16)²

= √(16^(⅔))

=256^(⅓)

= 4∛4

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Answer:

C.

Step-by-step explanation:

Took the test. :)

The two plants would have the same height in 7 weeks and the height would be 31 inches.

The form of the expression that represents the length of the plant on the window sill is:

initial length + (length of growth per week x number of weeks)

3 + (4 x w)

3 + 4w

The form of the expression that represents the length of the plant on the coffee table is:

initial length + (length of growth per week x number of weeks)

17 + (2 x w)

17 + 2w

When the two plants have the same height, the two expressions would be equal:

17 + 2w = 3 + 4w

17 - 3 = 4w - 2w

14 = 2w

w = 14/2

w = 7 weeks

Height of the plant = 3 + 4(7)

3 + 28 = 31 inches

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The measure of angle DBC is 60 degrees

Any triangle's three angles always sum to 180 degrees. Therefore, if you only have two angles available, add them up, and then take 180 away from the result.

The computation of the measure of angle ABC is shown below:

Given that

Angle ABC is 120 degrees

And, the measure of angle BAC is 60 degrees

So based on the above information

The  measure of angle  is

= Angle ABC - Angle ABD

= 120 degrees - 60 degrees

= 60 degrees

hence, the measure of angle DBC is 60 degrees

Q - Angle ABC is equal to 120 degrees. The measure of angle ABD is 60 degrees. What is the measure of angle DBC?

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The weight of Franklin’s mountain bike in kilograms which  weighs 17,000 grams is equal to option A. 17 kilograms.

Weight of Franklin’s mountain bike  in grams is equal to 17,000 grams

Convert the grams into kilograms.

One kilograms is equal to one thousand grams

1 kilograms = 1000 grams

⇒ 1 grams = ( 1 / 1000 ) kilograms

⇒ 17,000 grams = ( 17,000 / 1000 ) kilograms

⇒ 17,000 grams = 17 kilograms

Therefore, after converting grams into kilograms the weight of Franklin’s mountain bike whose weight in grams is 17,000 grams  is equal to option A. 17 kilograms.

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The length of the unknown side is 11.2ft

Using the Pythagorean theorem which states that the square of the hypotenuse side is equal to the sum of the squares of the other two sides.

From the image shown,we have the parameters as;

Now, substitute the values, we have;

x² = 10² + 5²

Find the squares

x² = 100 + 25

Add the values

x ² = 125

Find the square root

x = 11.2 ft

Hence, the value is 11.2ft

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Answer:

Step-by-step explanation:

Perpendicular line will have a slope that is the negative reciprocal so slope of 3 becomes - [tex]\frac{1}{3}[/tex].

then use point slope:

y - y = m(x - x)

Use the 2 points (8, -4)

y - -4 = -[tex]\frac{1}{3}[/tex] (x - 8)

y + 4 = -[tex]\frac{1}{3}[/tex] (x - 8)

y + 4 = -[tex]\frac{1}{3}[/tex]x + [tex]\frac{8}{3}[/tex]

y + 4 - 4 = -[tex]\frac{1}{3}[/tex]x  - [tex]\frac{4}{3}[/tex]

y = - [tex]\frac{1}{3}[/tex]x - [tex]\frac{4}{3}[/tex]

Parallelline will have the same slope m = 3.

use point slope again

y - -4 = 3(x -8)

y + 4 = 3x - 24

y + 4 -4 = 3x - 24 -4

y = 3x -28

The sides of the right triangle are as follows:

x = 18 units

y = 15.6 units

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Therefore, let's find the sides of the right triangle using trigonometric ratios

tan 30° = opposite / adjacent

tan 30° = 9 / y

cross multiply

y = 9 / tan 30

y = 15.5884645362

y = 15.6 units

Let's find x

sin 30 = opposite / hypotenuse

sin 30° = 9 / x

x = 9 / 0.5

x = 18 units

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The equation of the circle graphed above is equal to (x - 1)² + (y - 6)² = 9.

In Mathematics, the standard form of the equation of a circle is represented by this mathematical expression;

(x - h)² + (y - k)² = r²

Where:

By critically observing the graph of this circle, we have the following parameters:

Radius, r = 3 units.

Center, (h, k) = (1, 6).

Substituting the given parameters into the equation of a circle formula, we have the following;

(x - h)² + (y - k)² = r²

(x - 1)² + (y - 6)² = 3²

(x - 1)² + (y - 6)² = 9.

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The polynomial for the shaded area will be 3x² + 15x and x² + 6x + 8.

Let W be the rectangle's width and L its length.

The area of the rectangle is the multiplication of the two different sides of the rectangle. Then the rectangle's area will be

Area of the rectangle = L × W square units

19. The area is given as,

A = (x + 5)(3x)

A = 3x² + 15x

19. The area is given as,

A = (x + 4)(x + 2)

A = x² + 6x + 8

The polynomial for the shaded area will be 3x² + 15x and x² + 6x + 8.

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The additional piece of information needed to prove that ΔLKN ~ ΔKNM by the SAS similarity theorem is option C, MN/LK||MN.

The SAS (Side-Angle-Side) similarity theorem states that if two triangles have two pairs of corresponding sides that are in proportion, and the included angles are congruent, then the triangles are similar.

In this case, we know that ΔLKN and ΔKNM share the angle at vertex K, and we also know that LN/MN and LK/KN are in proportion.

However, we need to establish that MN/LK||MN, which means that MN is parallel to LK and forms a transversal with LN.

This information is necessary to establish that the included angles between the corresponding sides are congruent, which is required for the triangles to be similar.

Therefore, option C is the correct additional piece of information needed to prove that ΔLKN ~ ΔKNM by the SAS similarity theorem.

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Answer:

Step-by-step explanation:

6(x + 4)/5(x - 7) * 9(x - 7)/7(x + 4) =

Cancel the x + 4 and x - 7

54/35

The answer to the given fill in the blanks is "Residual". in a simple linear regression model we use the normal quantile plot of the residuals to evaluate if it is reasonable to assume the Residual come from a normal distribution.

In linear regression, we make several assumptions about the relationship between the predictor variable and the response variable. One of these assumptions is that the residuals (the differences between the predicted values and the actual values) follow a normal distribution with a mean of zero and constant variance. To check this assumption, we can create a normal quantile plot of the residuals. A normal quantile plot is a graphical method that compares the distribution of the residuals to a normal distribution. If the residuals are normally distributed, the plot should follow a straight line. If the plot deviates from a straight line, it suggests that the residuals are not normally distributed. A non-normal distribution of residuals can have several implications, such as indicating the presence of outliers or influencing the accuracy of the model’s prediction. Therefore, checking for normality of residuals is a crucial step in evaluating the model's assumptions and determining whether the model is appropriate for the data.

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£40 is shared as £5, £15 and £20. 88p is shared as 16p, 32p and 40p. 180° is shared as 40°, 40° and 100°

An equation is an expression that shows how numbers and variables are linked together using mathematical operations such as addition, subtraction, multiplication and division.

a) Share £40 in the ratio 1:3:4

For the first ratio = (1/8) * £40 = £5

For the second ratio = (3/8) * £40 = £15

For the third ratio = (4/8) * £40 = £20

b) Share 88p in the ratio 2:4:5

For the first ratio = (2/11) * 88p = 16p

For the second ratio = (4/11) * 88p = £32

For the third ratio = (5/11) * 88p = 40p

c) Share 180° in the ratio 2:2:5

For the first ratio = (2/9) * 180° = 40°

For the second ratio = (2/9) * 180° = 40°

For the third ratio = (5/9) * 180° = 100°

The ratios are shown above

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The next two days when Alicia would be allowed to eat junk foods are Thursday and Tuesday.

The first day after Saturday when Alicia would have junk food can be determined by counting the fifth day after Saturday.

Sunday - first day

Monday - second day

Tuesday - third day

Wednesday - fourth day

Thursday - fifth day

The next day after Thursday when Alicia would have junk food is the fifth day after Thursday.

Friday - first day

Saturday - second day

Sunday - third day

Monday - fourth day

Tuesday - fifth day

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Answer: To determine the amount in Heidi's 401(k) account after 15 years, we need to calculate her total contribution each month and the interest earned on that contribution over the 15-year period.

First, let's calculate Heidi's contribution each month:

11% of $3,250 = $357.50

Next, let's calculate her employer's contribution each month:

6% of $3,250 = $195

Since Heidi's contribution is $357.50, and her employer matches her contribution up to 6% of her salary, her employer will contribute $195 each month.

So, Heidi's total contribution each month is $357.50 + $195 = $552.50

Next, we can use the formula for compound interest to calculate the balance in her 401(k) account after 15 years:

A = P * (1 + r/n)^(nt)

where:

A is the balance in the account after t years

P is the principal (initial deposit)

r is the interest rate

n is the number of times the interest is compounded in a year

t is the number of years

In this case, n = 12 (because interest is compounded monthly) and t = 15 (because the calculation is for 15 years). The principal P is $552.50 (Heidi's total contribution each month). The interest rate r is 6.25%.

Plugging in the values, we get:

A = $552.50 * (1 + 0.0625/12)^(12 * 15) = $552.50 * (1.00520833)^180

Using a calculator or spreadsheet software, we can calculate this value to be approximately $27,973.81.

So, the amount in Heidi's 401(k) account after 15 years, rounded to the nearest cent, is $27,973.81.

Step-by-step explanation:

The following statements are true about the function f(x) = -( x-4) ² -1

1. The y intercept is (0,-17)

2. The axis of symmetry x = 4

3. The x intercept is (3,0)

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

The equation f(x) = -(x-4)² -1 can be expanded as

f(x) = -x²+8x-16-1 = -x²+8x-17

therefore when x = 0

the y-intercept = -17

therefore the y intercept = (0,-17)

The x intercept of the function,

when y = f(x) and y = 0

-(x-4)² = 1

-(x-4) = 1

-x+4 = 1

-x = -3

x = 3

the x intercept of the function is (3,0)

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The area of the side panel of the stairway is 360 square inches.

The side panel of the stairway is essentially a right triangle with the three steps forming the three horizontal segments (treads) and the three risers forming the three vertical segments (risers).

The height of the side panel would be the total height of the three risers, which is -

3 risers x 8 inches/riser = 24 inches

The length of the side panel would be the total length of the three treads, which is -

3 treads x 10 inches/tread = 30 inches

Now, we can calculate the area of the side panel using the formula for the area of a right triangle:

Area = (base x height)/2

we have the base as the length of the side panel (30 inches) and the height is the total height of the three risers (24 inches).

Substituting these values into the formula we get,

Area = (30 inches x 24 inches)/2

Area = 360 square inches

Therefore, the area of the side panel of the stairway is 360 square inches.

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The correct interpretation of the 90% confidence interval is given as follows:

There is a 90% chance that the true proportion of teenagers who drive their own car to school will be (0.5907, 0.9093).

It means that it is x% likely that the true population parameter(mean/proportion/standard deviation) is between a and b, which are the bounds of the confidence interval.

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Multivariable calculus in college is generally more challenging than AP Calculus BC due to the additional complexity of working with functions of several variables.

Multivariable calculus in college typically goes beyond the content covered in AP Calculus BC in high school. While both cover calculus concepts, multivariable calculus adds the complexity of working with functions of several variables, including partial derivatives, multiple integrals, and vector calculus. As a result, the difficulty level of multivariable calculus in college can be significantly higher than AP Calculus BC. However, this can vary depending on the individual student and the rigor of their high school AP course.

AP Calculus AB and BC are two levels of Advanced Placement calculus courses typically offered in high school.

AP Calculus AB covers a wide range of calculus topics including limits, derivatives, integrals, applications of derivatives and integrals, and the Fundamental Theorem of Calculus. The focus is on single-variable calculus, meaning students work with functions of one variable.

AP Calculus BC, on the other hand, covers all the topics included in AB as well as additional concepts such as parametric, polar, and vector functions, series, and sequences. The course is designed to be more challenging and covers more advanced topics in calculus, including applications of calculus in physics and engineering.

Overall, AP Calculus BC is considered to be a more rigorous course than AP Calculus AB, but both courses prepare students for college-level calculus.

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When the smaller data point is included, the mean will drop when compared to the other values, but the median may increase or decrease depending on whether the collection of numbers is even or odd.

1. Calculate the original set of numbers' mean and median.

2. Include the lesser data point in the collection.

3. Determine the updated median and mean.

4. Evaluate the original mean and median in comparison to the new mean and median. Depending on whether the collection of numbers is even or odd, the median will decline while the mean will increase.

We must first determine the mean and median of the initial set of data in order to evaluate how the addition of a smaller data point will impact those values. Adding together all the values and dividing by the total number of values yields the mean. We place the middle value in the set of integers after sorting them from least to largest to determine the median. We include the new, smaller data point in the set after finding the mean and median. We next perform the same calculations as previously to determine the new mean and median. Finally, we contrast the updated mean and median with the baseline values. The median may stay the same or drop depending on whether the set of numbers is even or odd, while the mean often decreases as it is compared to other values. The mean and median of a set of numbers can therefore be significantly affected by the addition of a tiny data point.

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The complete question is

how do you know how adding a smaller data point will affect the mean or median if its smaller than them other value.

Increase the accuracy of your measurements by using a more precise instrument or tool, such as a protractor or ruler.

The accuracy of the intersection of your three circles may not have been exact due to imprecise measurements taken. To improve the results, you could use a more precise instrument or tool, such as a protractor or ruler. With a protractor , you can measure the angle of the intersection of the circles more accurately. If a ruler is used, make sure that it is a straight edge, and measure the distance between the circles to ensure they intersect at the same point . Once the measurement is taken, draw the circles and their intersection accordingly. Doing this will help improve the accuracy of the intersection of the three circles and yield better results.

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Answer:

$0.22.

Step-by-step explanation:

To find the cost per ounce of juice, divide the total cost of 32 ounces of juice by the number of ounces:

7.04 ÷ 32 ounces = 0.22 per ounce

So, the cost per ounce of juice is approximately $0.22.

Answer:

0.22

Step-by-step explanation:

Answer: Look below :)

Step-by-step explanation:

Hello again Sarah!

a. the dashed line indicated our answer will use > or <

furthermore, we can see two points, (0,2) and (1,6) on the graph.

So the slope is: 4/1 = 4

and the y intercept can be visually seen as 2

Since the shaded part is to the right, we will use <

The final equation is: y < 4x + 2

I wont show working for the others, but its pretty similar:

b. [tex]y \geq 0.5x - 3[/tex]

c. y < 0.75x - 2

The function [tex]f(x) = 4x^3 - 4[/tex] is a polynomial function. It takes an input value x and returns an output value f(x) based on the expression [tex]4x^3 - 4.[/tex]So the value of f(-1/2) is -4.5.

When we evaluate f(-1/2), we substitute -1/2 for x in the expression for f(x). This gives us:

[tex]f(-1/2) = 4(-1/2)^3 - 4[/tex]

The expression inside the parentheses, [tex](-1/2)^3,\\[/tex] is the cube of -1/2. To simplify this expression, we need to first raise -1/2 to the power of 3:

[tex](-1/2)^3 = (-1/2) * (-1/2) * (-1/2) = -1/8[/tex]

Next, we substitute this value back into the expression for f(-1/2):

f(-1/2) = 4(-1/8) - 4

Now we can simplify the expression by multiplying 4 and -1/8:

f(-1/2) = -1/2 - 4

Finally, we subtract 4 from -1/2 to get the final answer:

f(-1/2) = -4.5

So the value of f(-1/2) is -4.5.

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Given the function f(x) = 4x^3 - 4, find the value of f(- 1/2)?