Solve the right triangle

Answer:

D. (1, 3)

Step-by-step explanation:

The solution is the coordinates of the intersection of the 2 lines

It took about 4.04 years for the number of students to reach 365.

The correct option is D.

The logarithm is the opposite of exponentiation. This shows that the logarithm of x to the base of b is the exponent to which b must be raised in order to reach the number x.

Given:

There were 240 students enrolled in Birchwood Elementary School during its first academic year.

The number of students grew significantly throughout the course of each of the ensuing ten years.

We may use logarithms to find n, the number of years.

By multiplying both sides of the equation by their logarithms, we obtain:

log(240(1.15)ⁿ) = log(365)

Using the rule of logarithms that says log(ab) = log(a) + log(b) and the fact that log(1.15) is a constant,

we can simplify the left side of the equation:

log(240) + nlog(1.15) = log(365)

Subtracting log(240) from both sides, we get:

nlog(1.15) = log(365) - log(240)

Using the rule of logarithms that says log(a/b) = log(a) - log(b), we can simplify the right side of the equation:

nlog(1.15) = log(365/240)

nlog(1.15) = log(1.52083)

Dividing both sides by log(1.15), we get:

n = log(1.52083) / log(1.15)

n ≈ 4.04

Therefore, n = 4 years.

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By using distributive property, an estimate for sin((π/6 + ε)(1 - ε)) is approximately 0.5108.

What is sinx ?

sinx is a trigonometric function that represents the ratio of the length of the side opposite to an angle x in a right-angled triangle to the length of the hypotenuse of the triangle. In other words, it is the sine of the angle x.

The sine function is typically denoted as sin(x), where x is the measure of an angle in radians. It is defined as follows:

sin(x) = opposite side / hypotenuse

According to the question:
We can use the small angle approximation sin(x) ≈ x for small angles x to estimate[tex]sin((\pi /6 + \epsilon)(1 - \epsilon))[/tex], where ε = 0.01.

First, we can expand the expression using the distributive property:

[tex]sin((\pi /6 + \epsilon)(1 - \epsilon)) = sin(\pi /6 - \epsilon^2 +\epsilon /6)[/tex]

Since ε is small, we can use the small angle approximation to estimate the value of sin(ε/6):

[tex]sin(\epsilon /6) =\epsilon/6[/tex]

Next, we can use the fact that [tex]sin(\pi /6) = 1/2[/tex] to estimate [tex]sin(\pi /6 - \epsilon ^2)[/tex] :

[tex]sin(\pi /6 - \epsilon ^2) ≈ sin(\pi /6) = 1/2[/tex]

Substituting these approximations into the original expression, we get:

[tex]sin((\pi /6 + \epsilon)(1 - \epsilon)) =sin(\pi /6 - \epsilon^{2} + \epsilon/6) =sin(\pi /6) + sin(\epsilon/6) - sin(\epsilon^{2} )sin((\pi /6 + \epsilon)(1 - \epsilon)) = 1/2 + \epsilon/6 - sin(\epsilon^{2} )[/tex]

Finally, we can use the fact that sin(x) is approximately equal to x for small values of x to approximate [tex]sin(\epsilon^{2} ) as \epsilon^{2}[/tex]:

[tex]sin(\epsilon^2) = \epsilon^2[/tex]

Substituting this approximation into the previous expression, we get:

[tex]sin((\pi /6 + \epsilon)(1 - \epsilon)) ≈ 1/2 + \epsilon/6 - \epsilon^2[/tex]

Now, we can substitute the value of ε = 0.01 into this expression to get an estimate:

[tex]sin((\pi /6 + \epsilon)(1 - \epsilon)) ≈ 1/2 + 0.01/6 - 0.01^2[/tex]

[tex]sin((\pi /6 + \epsilon)(1 - \epsilon)) =0.5108[/tex]

Therefore, an estimate for sin((π/6 + ε)(1 - ε)) is approximately 0.5108.

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The measure of the length DC would be 28.9 m.

A triangle is a two - dimensional figure with three sides and three angles.

The sum of the angles of the triangle is equal to 180 degrees.

∠A + ∠B + ∠C = 180°

Given is that ABC is right angle triangle with horizontal ground AD.

We can write -

DC = 100 - 82 sin (60)

DC = 100 - 82 x √3/2

DC = 100 - 71.1

DC = 28.9 m

Therefore, the measure of the length DC would be 28.9 m.

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Step-by-step explanation:

We can solve for x using the properties of logarithms as follows:

log2 (5x - 2) = -2

2^(-2) = 5x - 2 (using the definition of logarithms)

1/4 = 5x - 2 (simplifying 2^(-2))

5x = 2 + 1/4 = 9/4

x = 9/20

Therefore, the solution to the equation log2 (5x - 2) = -2 is x = 9/20.

Answer:

Using the formula for the number of months to pay off a credit card balance with minimum payments:

Number of months = (-1/30) * log(1 - (balance * monthly rate / payment))

where balance is the initial balance, monthly rate is the APR divided by 12, and payment is the minimum payment.

Plugging in the given values, we get:

Number of months = (-1/30) * log(1 - (500 * 0.01824/12 / 10))

Number of months = 46.05

Rounding up to the nearest whole number, it will take Ramiro 47 months to pay off his credit card balance with only minimum payments.

Step-by-step explanation:

Each piece of pizza has an area of  4.08π square inches.

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.

The area of a circle with radius r is given by the formula A = πr².

The pizza has a radius of 7 inches

Area of pizza is A = π(7²) = 49π square inches.

The pizza is cut into 12 equal-sized pieces

which means each piece represents 1/12th of the total area of the pizza.

Therefore, the area of each piece is:

(1/12) × 49π = 4.08π square inches

Hence, each piece of pizza has an area of 4.08π square inches.

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The magnitude of resultant of the two forces is 104pounds and angle of resultant is 37°

The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R.

using parallelogram law of vectors

R² = 64² +68²+2(64)(68)cos 76

R² = 8720 + 8704cos 76

R² = 8720+2105.7

R² = 10825.7

R = √ 10825.7

R = 104pounds( nearest pounds)

represent resultant angle by x

using sine rule

64/sinX = 104/sin(180-76)

64/sinX = 104/sin104

104sinX = 64sin 104

104sinX = 62.1

sinX = 62.1/104

sinX = 0.597

X = sin^-1(0.597)

X = 37°( nearest degree)

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Answer:

No, Sasha's conclusion is not correct. If her last line is 0 = 13, this means that 0 is equal to 13, which is not true. Therefore, there is no solution to the equation that Sasha solved.

It's possible that Sasha made an error in her algebraic manipulations that led to this incorrect conclusion. It's also possible that there is no solution to the equation. In either case, the fact that her last line is 0 = 13 indicates that there is an error in her solution or that the equation is inconsistent and has no solution.

Step-by-step explanation:

step-by-step explanation of why Sasha's conclusion is not correct:

Given: Sasha's last line of her solution is 0 = 13.

Recall that the goal of solving an equation is to find the value(s) of the variable that make the equation true.

If the equation is true, then both sides of the equation should be equal.

However, Sasha's last line states that 0 is equal to 13. This is not true, since 0 is not equal to 13.

Therefore, the conclusion that the solution to the equation is 13 is not correct.

It's possible that Sasha made an error in her algebraic manipulations, which led to this incorrect conclusion.

Alternatively, it's possible that the equation itself is inconsistent and has no solution.

In either case, the fact that Sasha's last line is 0 = 13 indicates that there is an error in her solution or that the equation is inconsistent and has no solution.

No, Sasha's conclusion is not correct. If her last line is 0 = 13, this means that 0 is equal to 13, which is not true. Therefore, there is no solution to the equation that Sasha solved.

It's possible that Sasha made an error in her algebraic manipulations that led to this incorrect conclusion. It's also possible that there is no solution to the equation. In either case, the fact that her last line is 0 = 13 indicates that there is an error in her solution or that the equation is inconsistent and has no solution.

Given: Sasha's last line of her solution is 0 = 13.

Recall that the goal of solving an equation is to find the value(s) of the variable that make the equation true.

If the equation is true, then both sides of the equation should be equal.

However, Sasha's last line states that 0 is equal to 13. This is not true, since 0 is not equal to 13.

Therefore, the conclusion that the solution to the equation is 13 is not correct.

It's possible that Sasha made an error in her algebraic manipulations, which led to this incorrect conclusion.

Alternatively, it's possible that the equation itself is inconsistent and has no solution.

In either case, the fact that Sasha's last line is 0 = 13 indicates that there is an error in her solution or that the equation is inconsistent and has no solution.

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Answer:

The correct answer that you are looking for is the following equations make x = 2 correct: x2 - 4 = 0, 4x2 = 16, and 2(x-2)2 = 0

The following equations make x = -2 correct: x2 = -4, and 3x2+12=0

Step-by-step explanation:

When finding out if x fits a certain number, there are many things to keep in mind, such as making sure that both sides are equal, making that if x changes the equation is resolved correctly, and also having to solve all the equations, of course.

The strategy that I chose was replacing x with either 2, or -2, and whichever one matched the final result would mean that x does equal that term.

Another strategy that could be more beneficial is solving the equation with step-by-step ideas.

I replaced x as 2 in x2 - 4 = 0, 4x2 = 16, and 2(x-2)2 = 0 and they matched the result.

For the remaining equations that didn't match the result I replaced x with -2 and then they finally matched.

I replaced x as -2 in x2 = -4, and 3x2+12=0 and they matched the result.

If in any shape or form, I was incorrect please inform me and I'll attempt to resolve the issue and see my mistake, sorry if thi was wordy and long.

I hope this was helpful!

The area of the garden alone is 21 yd²

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

The total length of the patio and garden is 6 yards while the total width is 5 yard. Hence:

Area of garden and patio = length * width = 6 yards * 5 yards = 30 yd²

The length of patio is 3 yard and width of the patio is 3 yard, hence:

Area of Patio = length * width = 3 yd. * 3 yd. = 9 yd²

Therefore:

Area of garden = Area of garden and patio - Area of Patio = 30 yd² - 9 yd² = 21 yd²

The area of the garden is 21 yd²

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The area of the rhombus is twice the area of one triangle:

A_rhombus = 2 * A

A_rhombus = 2 * (21/4)√10

A_rhombus = 21/2 * √10

A form's area is determined using the surface. To calculate the area of a rectangle or square, multiply its length and width. A is x times y in size.

To find the area of the rhombus, we can decompose it into two congruent triangles and find the area of one triangle, then multiply it by 2 to get the area of the whole rhombus.

The height of each triangle is h = 7 units, and the length of the base of each triangle is half the length of a diagonal of the rhombus. Using the Pythagorean theorem, we can find the length of the diagonal:

d² = x² + y²

d² = 9² + 3²

d² = 90

d = √90 = 3√10

Therefore, the length of the base of each triangle is 1/2 * 3√10 = (3/2)√10 units.

The area of one triangle is:

A = (1/2) * base * height

A = (1/2) * (3/2)√10 * 7

A = (21/4)√10

The area of the rhombus is twice the area of one triangle:

A_rhombus = 2 * A

A_rhombus = 2 * (21/4)√10

A_rhombus = 21/2 * √10

Therefore, the area of the rhombus is 21/2 * √10 square units.

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Answer: 6x+6

Step-by-step explanation: The equation we will be using here is Perimeter of rectangle = 2*(Length+Width) We are given that the width is 3x-2 and the length is 5. We can plug into the perimeter equation to get Perimeter = 2*(3x-2+5).

2*(3x-2+5)

2*(3x+3)

6x+6  Distributive property

Our perimeter is 6x+6 in terms of x.

Hope this helps!

[tex] \Large{\boxed{\sf Height = 58 \: m}} [/tex]

[tex] \\ [/tex]

The volume of a cylinder can be calculated using the following formula:

[tex] \Large{\sf V = B \times h} [/tex]

Where:

[tex] \\ [/tex]

Since we are not given the area of its base, which is a circle, we will have to calculate it applying the following formula:

[tex] \Large{\sf B = \pi \times r^2} [/tex]

Where:

[tex] \\ [/tex]

Combining these two formulas, we can express the volume of the cylinder as follows:

[tex] \Large{\sf V = \underbrace{\sf \pi \times r^2}_{B} \times h} [/tex]

[tex] \\ [/tex]

Now, rearrange the formula to isolate the height, h.

[tex] \sf V = \pi \times r^2 \times h \Longleftrightarrow \dfrac{V}{\pi \times \times r^2} = \dfrac{ \pi \times r^2 \times h}{\pi \times r^2 } \Longleftrightarrow h = \dfrac{V}{\pi \times r^2} [/tex]

[tex] \\ [/tex]

[tex] \Large{\boxed{\sf Given \text{:} } \begin{cases} \sf V &=\sf 492,452.48 \: m^3 \\ \sf r &=\sf 52 \: m \: \\ \sf \pi &= \sf 3.14 \: (approximately) \end{cases} } [/tex]

[tex] \\ [/tex]

Substitute these values into our formula:

[tex] \sf h = \dfrac{492,452.48 \: m^3}{(52 \: m)^2 \times 3.14 } = \dfrac{492,452.48 \: m^3}{8,490.56\: m^2 } \\ \\ \\ \\ \implies \boxed{\boxed{\sf h = 58 \: m}} [/tex]

Set D ∈ {76, 90, 92, 94, 96}

Sets are groups of clearly defined objects or elements in mathematics. A set is denoted by a capital letter, and the cardinal number of a set is enclosed in a curly bracket to indicate how many members there are in a finite set.

As per the given data:

Set D consists of positive two-digit even numbers greater than 74 without the digit 8 in them.

For writing set D in the roster form:

For greater than 74:

{76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98}

Without the digit 8:

{76, 90, 92, 94, 96}

Hence, Set D ∈ {76, 90, 92, 94, 96}

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Answer:

the slope is 24/145

Step-by-step explanation:

slope= rise/run

48/290

24/145

the slope is 24/145

The area of the resulting figure is 4 times smaller than the area of the original figure.

And, When the side lengths are multiplied by 2, the perimeter is 2 times the original perimeter.

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. And, Opposite sides of the rectangle are equal and parallel to each other.

We know that;

When two figures are similar, then ratio of its areas is also equal to the scale factor squared.

Hence, We can assume that the figure is a rectangle or a triangle;

Let

z = the scale factor

x = Area of the resulting figure

y = Area of the original figure

So, We get;

z² = x/y

Since, we have;

z = 1/2

Hence, We can substitute;

⇒ (1/2)² = x/y

⇒ x/y = 1/4

⇒ x = y/4

Therefore, The area of the resulting figure is 4 times smaller than the area of the original figure.

And, If the side lengths are multiplied by 2.

We get;

Perimeter = 2 (L + W)

                = 2 (2L + 2W)

                = 2 (2L + 2W)

Thus, When the side lengths are multiplied by 2, the perimeter is 2 times the original perimeter.

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Answer:42

Step-by-step explanation: 42x0=42

The surface area of Neelah's package is 214 square inches.

What is surface area, and how is it calculated?

The surface area is the measure of the total area that the surface of an object occupies. It is calculated by adding up the areas of each face or surface of the object.

Calculation of the surface area:

To find the surface area of Neelah's package, we first need to determine the dimensions of the package. We are given that the volume of the package is 748 cubic inches and that the equation 4 x 11 x h = 748 can be used to find the height of the package.

Solving for h, we have:

4 x 11 x h = 748

44h = 748

h = 17

Therefore, the dimensions of the package are 4 inches by 11 inches by 17 inches.

To find the surface area of the package, we need to add up the area of each face of the package. The package has six faces, so we calculate the area of each face as follows:

Front and back faces: 4 inches x 17 inches = 68 square inches each

Top and bottom faces: 11 inches x 17 inches = 187 square inches each

Left and right faces: 4 inches x 11 inches = 44 square inches each

Therefore, the total surface area of the package is:

2(68) + 2(187) + 2(44) = 136 + 374 + 88 = 598 square inches

However, this calculation includes the interior of the package, and we are only interested in the external surface area. The top and bottom faces are not part of the external surface area, so we need to subtract them from the total:

598 - 2(187) = 224 square inches

Therefore, the surface area of Neelah's package is 214 square inches.

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Answer:

4.4 billion years

Step-by-step explanation:

The decay of Uranium-238 to lead is a first-order radioactive decay process, and the amount of remaining Uranium-238 after time t can be modeled by the exponential decay formula:N(t) = N0 * e^(-λt)where N0 is the initial amount of Uranium-238, N(t) is the remaining amount after time t, and λ is the decay constant.The half-life of Uranium-238 is 4.468 billion years, which means that the decay constant can be calculated as:λ = ln(2) / t1/2 = ln(2) / (4.468 * 10^9 years) ≈ 1.55125 x 10^-10 years^-1We are given that 49.7% of the original Uranium-238 remains, which means that 50.3% has decayed. Therefore, the ratio of remaining Uranium-238 to original Uranium-238 is:N(t) / N0 = 0.497Taking the natural logarithm of both sides of the equation and solving for t, we get:ln(N(t) / N0) = -λtt = -ln(N(t) / N0) / λPlugging in the given values, we get:t = -ln(0.497) / (1.55125 x 10^-10 years^-1) ≈ 4.40 billion yearsTherefore, the age of the Moon rock is approximately 4.40 billion years. Note that we should round the answer to two significant digits because the given data only has two significant digits. So, the final answer is:t ≈ 4.4 billion years.

Answer:i cant tell upside down

Step-by-step explanation:

so like huh

From the given polygon, The measure of angle K is 115 degrees.

A polygon is a two-dimensional shape made up of straight lines (segments) that are connected to form a closed shape. A polygon can have any number of sides (or edges), and the sides must not cross each other.

We know that the sum of the interior angles of a triangle is 180 degrees. We can use this fact to find the measure of angle J, which is opposite to angle K:

Angle J = 180 - 70 - 25 = 85 degrees

Now, we can use the fact that the sum of the interior angles of a quadrilateral is 360 degrees.

Angle K + 85 + 90 + 70 = 360

Simplifying the equation:

Angle K = 360 - 85 - 90 - 70 = 115 degrees

Therefore, the measure of angle K is 115 degrees.

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The range here is 50 & the midrange is 25.

Tο find the range, we need tο subtract the smaIIest number frοm the Iargest number in the set. The Iargest number in the set is 65 and the smaIIest number is 15,

Finding οut range:

Range = Biggest number - smallest number, which equals 65 - 15 to make 50.

Tο find the midrange, we need tο add the smaIIest number tο the Iargest number and divide by 2. The smaIIest number is 15 and the Iargest number is 65, sο the midrange is:

Midrange = (SmaIIest number + Largest number) / 2 = (15 + 65) / 2 = 40

Therefοre, the cοrrect οptiοn is The range is 50 and the midrange is 25.

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Complete question:

Which οf the statements beIοw is true fοr the fοIIοwing set οf numbers?

42, 15, 36, 51, 65, 28:

a) The range here is 50 and the midrange is 25.

b) The range here  is 50 and the midrange is 40.

c) The range here  is 14 and the midrange is 35.

d) The range here is 65 and the midrange is 39.

The cardinality of B and (A ⋂ C) will be n(B) = 3 and n(A ⋂ C) = 1.

In mathematics, cardinality refers to the size or number of elements in a set. It is a concept used to compare the sizes of sets, regardless of their specific elements. For instance, the cardinality of the set {1, 2, 3} is 3, while the cardinality of the set {2, 4, 6, 8, 10} is 5. Cardinality is typically denoted using vertical bars, such that |S| represents the cardinality of set S.

Now,

The symbol "n(S)" represents the cardinality

Given sets:

A = {1, 2, 3, 4, 5}

B = {1, 3, 5}

C = {4, 6}

To find the cardinality of the sets:

n(B) = 3

Set B has three elements: 1, 3, and 5.

n(A ⋂ C) = 1

The intersection of sets A and C is the set of elements that are in both A and C. Since A and C only have one element in common (the number 4), the cardinality of the intersection set A ⋂ C is 1. Therefore, n(A ⋂ C) = 1.

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A roll has 12 yards of ribbon. Haley uses 6 feet of the ribbon for a swing project. There are 30 feet of ribbon are left on the roll.

The foot is a unit of measurement. Length is measured by feet.

First, confirm that the numbers use the same measuring units. We can see that Haley used 6 feet out of the roll's 12 yards.

Three feet make up a yard.

Because 6 is equivalent to twice as much as 3 feet and 2 is equal to twice as much as 1, 6 feet would be equal to 2 yards.

So, we must take away 2 from 12; 12-2 equals 10.

Recall that the question asks for feet, therefore we may multiply 10 by 3 to get how many feet are still available.

10 x 3 = 30,

So, 30 feet of ribbon is left on the roll.

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Answer:

Step-by-step explanation:

To find the new population, we can substitute p = 4300 into the formula:

N = p + 0.04p

N = 4300 + 0.04(4300)

N = 4300 + 172

N = 4472

Therefore, the new population of the town is 4472 persons.

Answer:

74

Step-by-step explanation:

-6(-5-4) - 4(-3-2)

30 + 24 + 12 + 8

= 54 + 12 + 8

= 66 + 8

= 74

Answer:

Step-by-step explanation:

A- (-3,4)

left on the x-axis would result in negative 3

4 units above the x-axis would result in postive 4