What is the value of x in both these questions

The length of the missing side is given as follows:

[tex]x = \sqrt{155}[/tex]

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

For the triangle in this problem, the sides and the hypotenuse are given as follows:

Hence the missing side is given as follows:

x² + 13² = 18²

x = sqrt(18² - 13²)

x = sqrt(155) -> most simple radical form.

The triangle is given by the image presented at the end of the answer.

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To create point R such that a line through points P and R is parallel to line l, Gerald needs to draw a line through points T and N.

We have,

We know that line l is parallel to line m since both lines intersect at point Q and no other point.

The circle centered at Q intersects line m at point S and line l at point N, which means that QS is perpendicular to line l.

Similarly, the circle centered at P intersects line m at point T, which means PT is perpendicular to line l.

To create a line parallel to line l, Gerald needs to find a line perpendicular to line l.

This can be achieved by drawing a line through points T and N. This new line will be parallel to QS and perpendicular to line l.

Finally, Gerald can find the intersection of this new line with the circle centered at P to find point R, which will be equidistant from points P and T. Drawing a line through points P and R will be parallel to line l.

Thus,

To create point R such that a line through points P and R is parallel to line l, Gerald needs to draw a line through points T and N.

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Answer: it will take approximately 14.62 years for the initial investment to double at an interest rate of 4.75% per year, compounded continuously.

Step-by-step explanation: Given an investment of $2000 at a continuously compounded interest rate of 4.75%, the balance in the account can be calculated using the following mathematical expression after t years:

The aforementioned equation, A = P * e^(rt), denotes the relationship between the accrued amount (A) and the principal amount (P), compounded continuously at a fixed annual rate of interest (r) over a specific time period (t), as governed by the mathematical constant "e."

In the context of financial calculations, the symbol 'P' denotes the initial capital investment. The interest rate, represented by the variable 'r', is expressed in the form of a decimal. Additionally, 'e' is the mathematical constant, roughly equivalent to 2.71828. Finally, 't' refers to the duration of the investment, measured in years.

In order to determine the duration of time required for the investment to achieve a twofold increase, it is necessary to solve the corresponding equation:

The equation expressed as 2P = P * e^(rt) can be restated more formally as follows. Given a principal investment amount represented by P and a rate of return indicated by r, compounded over time t, the equation can be expressed as the product of P and the exponential function of e^(rt), yielding twice the initial investment amount.

The variable 2P represents the monetary value acquired through doubling the initial investment.

Upon division of both sides by P, the resulting expression is as follows:

The equation 2 equals the exponential function of the base e raised to the power of the product of r and t.

By applying the natural logarithm function to both expressions, the resultant outcome is:

The natural logarithm of 2 can be represented as rt, where r denotes the logarithmic base and t denotes the logarithm of the argument, in accordance with the conventions of academic mathematical writing.

Upon resolving for the variable t, an outcome is yielded:

The mathematical expression t = ln(2) / r can be written in a formal academic style as follows: The equation determines the relationship between time t and the rate of decay r, where t is equal to the natural logarithm of 2 divided by r.

Upon substitution of the provided values, the resultant output is:

The calculated value of the variable t, representing the length of time in years, is approximately equal to 14.62 years, obtained through the algebraic manipulation of the natural logarithmic function of 2 divided by the constant value of 0.0475.

Answer:

see below

Step-by-step explanation:

L = 2W-6

given: (L+3)*(W+3) = 270

so (2W-6+6)*(W+6) = 270

so 2W*(W+6) = 270

open up the brackets

2w²+12W = 270

so solve quadratic equations of 2w²+12W - 270 =0

w = 9 or -15

so dimensions of the unframed photograph = width 9, length 12

Answer:

Step-by-step explanation:

The possible outcomes for the coin toss being tails and the card chosen being 1, 2, or 3 are:

-T1

-T2

-T3

Therefore, there are three possible outcomes for the event that the coin toss is tails.

Answer:

The method that could be used to fairly choose the class president in the given scenario is flipping a fair coin. This is because flipping a fair coin is a random process and both candidates have an equal chance of winning. This ensures fairness in the selection process. Asking a group of senior teachers or letting the two candidates mutually decide who should win may introduce biases and may not be fair to both candidates. Rolling a biased die may also introduce unfairness as the outcome is not random and is predetermined.

Answer:

x = -16/7 and y = 38/7.

Step-by-step explanation:

To solve the system of equations:

x + 3y - 22 = 0 --- equation (1)

2x - y + 42 = 0 --- equation (2)

x - 11y + 142 = 0 --- equation (3)

We will use the method of substitution to find the values of x and y that satisfy all three equations.

From equation (1), we can express x in terms of y:

x = 22 - 3y --- equation (4)

We can substitute equation (4) into equation (2) and simplify:

2(22 - 3y) - y + 42 = 0

44 - 6y - y + 42 = 0

-7y = -38

y = 38/7

Now, we can substitute the value of y into equation (4) to find x:

x = 22 - 3(38/7)

x = -16/7

Therefore, the solution to the system of equations is:

x = -16/7 and y = 38/7.

Step-by-step explanation:

When we simplify the expression -5/9 - (-9/4), we can rewrite it as:

-5/9 + 9/4

To add these fractions, we need to find a common denominator. The least common multiple of 9 and 4 is 36, so we can convert both fractions to have a denominator of 36:

-5/9 = -20/36

9/4 = 81/36

Now we can substitute these values into our expression and add them:

-20/36 + 81/36 = 61/36

Therefore, the simplified expression is 61/36.

The height of the Ferris wheel at point A is 100 ft.

The height of the Ferris wheel at  point B is 182.1 ft.

The height of the Ferris wheel at  point C is 39.9 ft.

The height of the Ferris wheel at each point is calculated as follows;

The height of a point on a circle; H = h + y

Where;

H = h + rsinθ

where;

For point A with θ = 0 radians;

H = 100 + 85 x sin (0)

H = 100 ft

For point B with θ = 7π/12 radians;

H = 100  + 85 x sin(7π/12)

H = 182.1 ft

For point C with θ = 5π/4 radians;

H = 100  + 85 x sin(5π/4)

H = 39.9 ft

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The volume of the Triangular prism above is 156 cm^3

What is a Triangular Prism?

Triangular Prism is a three-dimensional shape consisting of two triangular ends connected by three rectangles. It has two identical triangles as parallel bases and if it is a right prism all lateral faces are rectangles.

How to determine this

The volume of a Triangular prism is calculated as

Volume = 1/2 * base * height * length

Where the Base = 8 cm

Height = 3 cm

Length = 13 cm

Volume = 1/2 * 8 * 3 *13

Volume = 1/2 *312

Volume = 156 cm^3

Therefore, the volume  of the triangular prism is 156 cm^3

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keeping in mind that twin sides make twin angles, Check the picture below.

The answer is D :)  )

The range of laptops that they should produce per day to make profit is: 1 to 21 laptops

When dealing with inequalities, we could make use of any of the following:

Greater than(>)

Less than (<)

Greater than or equal to (≥)

Less than or equal to (≤)

We are given:

Cost function: C(x) = 4x² - 10x + 150

Revenue function: R(x) = -3x² + 150x + 75

Now, if they want to make profit, then:

R(x) > C(x)

Thus:

4x² - 10x + 150 > -3x² + 150x + 75

Rearranging to get:

7x² - 160x + 75 > 0

Solving using quadratic calculator gives:

x ≈ 1 and 21

Thus the range of laptops that they should produce per day to make profit 1 to 21

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The concentration of H⁺ ions at a pH of 7 is 1 x 10⁻⁷ M.

Ion concentration refers to the amount of ions that are present in a solution or a medium, expressed in terms of their concentration.

The concentration of ion with pH of 7 is calculated as follows;

pH = -log[H⁺]

[H⁺] = 10^(-pH)

The given pH = 7, so the ion concentration is calculated as;

[H⁺] = 10⁻⁷

[H⁺]  = 1 x 10⁻⁷ M

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Here is an attempt at an appropriately SFW 1 minute long rap about why the kite shape is the best:

The kite, soarin' high up in the sky,

Spreading colorful fabric far and wide.

There's no shape more perfect, this I must insist,

As it dances on the wind, ever blissful and lightly kissed.

Kite strings set it free, without any qualms,

Allowing it to float on high, away from harms.

Circling birds and beasts with graceful ease,

As a peaceful, wandering seas.

The kite shape is light but strong too,

Built to capture the breeze and soar true.

Diamond, square and box they can't compare,

To the kite so bright, beyond compare.

Many sizes to fit every quest,

From small pocket kites to giants they test.

Traditional or modern, simple or complex,

The kite is a form with many flexes.

Dragons and fighters, parafoils that stand,

On the beach or the park, across every land.

A kite for each mood and gusty seal,

As it dances and drifts from keel to keel.

So fly your kite up into the sun,

Some joyful moments, for now's the one.

Remember how free your kite did roam,

As a bird in the wind, and no longer alone.

The kite shape is art with physical grace,

Worth admiring for form, color and space.

There's magic up there among the open air,

As your kite soars high, beyond compare.

Here is an attempt at an appropriately SFW 1 minute long rap about why the kite shape is the best:

The kite, soarin' high up in the sky,

Spreading colorful fabric far and wide.

There's no shape more perfect, this I must insist,

As it dances on the wind, ever blissful and lightly kissed.

Kite strings set it free, without any qualms,

Allowing it to float on high, away from harms.

Circling birds and beasts with graceful ease,

As a peaceful, wandering seas.

The kite shape is light but strong too,

Built to capture the breeze and soar true.

Diamond, square and box they can't compare,

To the kite so bright, beyond compare.

Many sizes to fit every quest,

From small pocket kites to giants they test.

Traditional or modern, simple or complex,

The kite is a form with many flexes.

Dragons and fighters, parafoils that stand,

On the beach or the park, across every land.

A kite for each mood and gusty seal,

As it dances and drifts from keel to keel.

So fly your kite up into the sun,

Some joyful moments, for now's the one.

Remember how free your kite did roam,

As a bird in the wind, and no longer alone.

The kite shape is art with physical grace,

Worth admiring for form, color and space.

There's magic up there among the open air,

As your kite soars high, beyond compare.

Answer:

Step-by-step explanation:

Sin O= opp/hyp

Sin 55 = BC/AB

Sin 55=.82

so

BC/AB=.82

Let's start by setting up some equations to represent the given information.

Let's say the price of a postcard is "x" cents.

Then, according to the problem statement, the price of an envelope is 8 times cheaper than the notepad, which means the price of an envelope is (1/8)th of the price of the notepad. So the price of an envelope is (1/8)*y cents, where "y" is the price of the notepad in cents.

Also, we know that the price of an envelope is 2 cents more expensive than the price of a postcard, so we can write:

(1/8)*y = x + 2 ...(Equation 1)

We also know that there are 15 postcards and 10 envelopes in the purchase, so the total cost of the postcards and envelopes is:

15x + 10[(1/8)*y] ...(Equation 2)

Finally, we have a notepad that costs "y" cents. So the total cost of the purchase is:

15x + 10[(1/8)*y] + y ...(Equation 3)

We are given that the total cost of the purchase is 1 dollar and 68 cents, which is equal to 168 cents. So we can write:

15x + 10[(1/8)*y] + y = 168 ...(Equation 4)

Now we have four equations (Equation 1, Equation 2, Equation 3, and Equation 4) with three variables (x, y, and 168). We can solve for x and y by using a system of equations.

From Equation 1, we can solve for y in terms of x:

(1/8)*y = x + 2

y = 8x + 16

Substituting this expression for y into Equations 2 and 3, we get:

15x + 10[(1/8)*y] = 15x + 10(8x + 16) = 160x + 160

15x + 10[(1/8)*y] + y = 15x + 8x + 16 = 23x + 16

Substituting these expressions into Equation 4, we get:

23x + 16 = 168

Solving for x, we get:

x = 6

Substituting this value for x into Equation 1, we can solve for y:

(1/8)*y = x + 2 = 6 + 2 = 8

y = 64

So the price of a postcard is 6 cents, the price of an envelope is (1/8)*64 + 2 = 10 cents, and the price of a notepad is 64 cents

If the "y-value" is decreasing at rate of 2 units per second, the rate at which "x" is changing when x=3/2 is -1/3 units per second.

The point moves along a curve having equation as : y = 2x² + 1; and

We know that y is decreasing at a rate of 2 units per second. We have to find the rate at which "x" is changing when x = 3/2,

To solve this problem, we differentiate, the curve equation,

So, taking the derivative of both sides of the equation with respect to time "t",

We get,

⇒ d/dt (y) = d/dt (2x² + 1),

⇒ dy/dt = (4x) × dx/dt,

We are given that dy/dt = -2 (since y is decreasing at a rate of 2 units per second), and we need to find "dx/dt" when x = 3/2,

Substituting "x = 3/2" and "dy/dt = -2",

We get,

⇒ -2 = 4×(3/2)(dx/dt),

⇒ -2 = (6)×(dx/dt),

⇒ dx/dt = -1/3,

Therefore, the rate at which "x" is changing when x = 3/2 is -1/3 units per second.

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The given question is incomplete, the complete question is
A point moves along a curve y=2x² + 1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x= 3/2 ?

Answer:

Equation:  250(1 + 0.028t) = 500

Answer:  36 years

Step-by-step explanation:

The equation we can use to solve this problem is the simple interest formula:

[tex]\boxed{A = P (1 + rt)}[/tex]

where:

Given the initial investment is $250 at an interest rate of 2.8%, and Edward wants to double his money:

Substite these values into the equation:

[tex]500=250(1+0.028t)[/tex]

Swap sides:

[tex]250(1+0.028t)=500[/tex]

Now solve for t:

[tex]\implies \dfrac{250(1+0.028t)}{250}=\dfrac{500}{250}[/tex]

[tex]\implies 1+0.028t=2[/tex]

[tex]\implies 1+0.028t-1=2-1[/tex]

[tex]\implies 0.028t=1[/tex]

[tex]\implies \dfrac{0.028t}{0.028}=\dfrac{1}{0.028}[/tex]

[tex]\implies t=35.71428571...[/tex]

Assuming the interest is applied annually on the anniversary of the account opening, it will take Edward 36 years to double his money with a 2.8% simple interest rate.

The amount(volume) of water that will be needed to fill the pool is 144 ft³.

Volume is the space occupied by a solid shape.

To calculate the amount(volume) of water that will be needed to fill the pool, we use the formula below

Formula:

Where:

From the diagram in the question,

Given:

Substitute these values into equation 1

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Null hypothesis: H0: p ≤ 0.56

Alternative hypothesis: H1: p > 0.56

We have,

Let p be the true proportion of people in the population who support stricter gun laws.

The null and alternative hypotheses in symbolic form for the claim that more than 56% of people support stricter gun laws are:

Null hypothesis: H0: p ≤ 0.56

Alternative hypothesis: H1: p > 0.56

Thus,

The null hypothesis states that the proportion of people in the population who support stricter gun laws is less than or equal to 56%, while the alternative hypothesis states that the proportion is greater than 56%.

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Step-by-step explanation:

Let X be the amount invested at 6%, Y be the amount invested at 9%, and Z be the amount invested at 13%.

From the problem, we know that:

X + Y + Z = 7600 ---(1) (the total amount invested is $7600)

0.06X + 0.09Y + 0.13Z = 818 ---(2) (the total income from the investments is $818)

Z = X + Y + 1200 ---(3) (the amount invested at 13% is $1200 more than the amounts invested at 6% and 9% combined)

We can use equation (3) to substitute for Z in equations (1) and (2), then solve for X and Y as follows:

X + Y + (X + Y + 1200) = 7600

2X + 2Y = 6400

X + Y = 3200

0.06X + 0.09Y + 0.13(X + Y + 1200) = 818

0.06X + 0.09Y + 0.13X + 0.13Y + 156 = 818

0.19X + 0.22Y = 662

Using the system of equations X + Y = 3200 and 0.19X + 0.22Y = 662, we can solve for X and Y to get:

X = 800

Y = 2400

Substituting back into equation (3), we get:

Z = X + Y + 1200 = 4400

Therefore, the amounts invested at 6%, 9%, and 13% were $800, $2400, and $4400 respectively.

the distance between line t and point D(-7, 0) is approximately 3.5 units.

what is distance  ?

Distance is a measure of the amount of space between two objects or points. It is usually measured in units such as meters, kilometers, miles, or feet. Distance can be either a scalar quantity

In the given question,

To find the distance between line t and point D(-7, 0), we need to first find the point on line t that is closest to point D.

We can use the formula for the distance between a point and a line in the coordinate plane:

distance = |Ax + By + C| / √(A^2 + B^2)

where Ax + By + C is the equation of the line in standard form, and (x, y) is the coordinates of the point.

In this case, the equation of line t is y = -x - 2, which we can rewrite in standard form as x + y + 2 = 0.

Using the formula above, we have:

distance = |x + y + 2| / √(1^2 + (-1)^2)

distance = |(-7) + 0 + 2| / √(2)

distance = 5 / √(2)

distance ≈ 3.5 (rounded to the nearest tenth)

Therefore, the distance between line t and point D(-7, 0) is approximately 3.5 units.

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The graph of the equation is linear

Given data ,

In physical education class, students get one sticker for each mile they walk and two stickers for each mile they run.

And , Jenny earned nine stickers

and completed seven miles

Based on the information provided, Jenny earned 9 stickers and completed 7 miles in physical education class. She receives one sticker for each mile she walks and two stickers for each mile she runs.

We can represent this information on a graph with the number of stickers earned on the vertical axis (y-axis) and the number of miles completed on the horizontal axis (x-axis).

Hence , the graph is solved

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The sample data provides sufficient evidence to conclude that the population proportion is greater than 0.17 at 1% level of significance.

One can define a null hypothesis as a declaration that postulates no noteworthy variance or association between two or more variables existing within a populace.

Usually utilized in statistical hypothesis testing, it evaluates if an observed effect or relation is statistically meaningful or arises purely by chance.

It often bears the insignia H0 and subject to an alternative proposition (Ha) signifying a significant outcome or connection instead. Whenever the null hypothesis gets dismissed, one may deduce that there are ample findings to sustain the alternate assertion's validity.

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Carmine's loan balance after two years would be approximately $13,827.72.

In mathematics, the term "loan" typically refers to a sum of money that is borrowed by one party from another with the agreement to repay it with interest over time. Loans are a common financial concept used in various areas of mathematics, such as finance, economics, and business.

"Compounded quarterly" refers to a method of calculating interest or investment growth where the interest is added to the principal and then reinvested or recalculated at the end of each quarter (every three months) within a given time period.

To calculate Carmine's loan balance after two years, we need to use the compound interest formula, which is given by:

A = P ×[tex](1+r/n)^{nt}[/tex]

Where:

A = the loan balance after t years

P = the initial principal amount (loan amount) = $11,500

r = annual interest rate = 8% or 0.08 (expressed as a decimal)

n = number of times interest is compounded per year = 4 (since it is compounded quarterly)

t = time in years = 2 (since Carmine made no payments for the first two years)

Putting the values into the formula, we get:

A = 11,500 × (1 + 0.08/4)⁸

A = 11,500 × (1 + 0.02)⁸

A = 11,500 × (1.02)⁸

A ≈ $13,827.72

So, Carmine's loan balance after two years would be approximately $13,827.72.

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Answer: V=1578.28 m³

Step-by-step explanation:

Volume is given as the formula

[tex]V=\frac{2}{3}\pi r^{3}[/tex]

r, radius, is the line from the center point to any end of the semi-sphere(that's what this shape is called) Here they show radius as

r=9.1

Substitute r into the formula

[tex]V=\frac{2}{3}\pi (9.1)^{3}[/tex]    plug into calculator

V=1578.275    round to hundredths means 3 digits after the decimal point

V=1578.28  the number after the 7 is 5 or greater so you round up.