he side lengths of a square increase at a rate of 2 cm/s. Find the rate at which the area of the square is increasing when the side length is 6 cm.
Provide your answer below:

x(t) = \(c_1e^1^4^t+c_2e^-^t\)  , y(t) = \(2c_1e^1^4^t-c_2e^-^t\)   is the general solution of the sytem of differention eqution dx/dt = 4x+ 5y , dy/dt = 10x + 9y .

given  dx/dt = 4x+ 5y  and dy/dt = 10x + 9y

X'(t) = \(\left[\begin{array}{ccc}4&5\\10&9\end{array}\right]\)  X

where X = \(\left[\begin{array}{ccc}x\\y\\\end{array}\right]\)

so A = \(\left[\begin{array}{ccc}4&5\\10&9\end{array}\right]\)

now we need to find eigen value of the matrix and the corresponding eigen vector.

| A- λI | = 0

so after equation the value to zero

λ = 14 and λ = -1

now for the  λ = 14 corresponding eigen vector = \(\left[\begin{array}{ccc}1\\2\\\end{array}\right]\)

for λ = -1 corresponding eigen vector =  \(\left[\begin{array}{ccc}1\\-1\\\end{array}\right]\)

So general equation is given by:

X(t) = \(c_1e^1^4^t\)  \(\left[\begin{array}{ccc}1\\2\\\end{array}\right]\)  +  \(c_2e^-^t\)   \(\left[\begin{array}{ccc}1\\-1\\\end{array}\right]\)

after solving this

x(t) = \(c_1e^1^4^t+c_2e^-^t\)

y(t) = \(2c_1e^1^4^t-c_2e^-^t\)

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Answer:

D) 6^5 · 3^5

Step-by-step explanation:

(6 · 3)^5 = 1889568

A) WRONG

(5 · (6 · 3)) = 90

B) WRONG

9^5 = 59049

C) WRONG

6 · 3^5 = 1458

D) CORRECT

6^5 · 3^5 = 1889568

The equation that has no solution will be ax + b = ax + c. Then the correct option is A.

In other words, the collection of all feasible values for the parameters that satisfy the specified mathematical equation is the convenient storage of the bunch of equations.

If a, b, and c are three different numbers.

Let's check all the options, then we have

A: ax + b = ax + c, the variable gets eliminated, then the equation has no Solution.

B: ax + b = ax + b, the variable gets eliminated and constant too, then the equation has infinitely many Solutions.

C: ax = bc + c, the variable dost not get eliminated, then the equation has a unique Solution.

The equation that has no solution will be ax + b = ax + c. Then the correct option is A.

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The complete question is given below.

If a, b, and c are three different numbers, which of the following equations has NO solutions?

A: ax + b = ax + c

B: ax + b = ax+b

C: ax = bc + c

Answer:

$17.45

Step-by-step explanation:

$17.45*3=52.35

$52.35+$45.99=$98.34

Answer:

(x-2)(x+2)

Step-by-step explanation:

use the difference of squares

x²-4 = x²-2² = (x-2)(x+2)

Answer:

the value of x in the equation is -52.

Step-by-step explanation:

To find the value of x in the equation:

(1/3)x - (2/3) = -18

We can start by isolating the variable x.

Add (2/3) to both sides of the equation:

(1/3)x = -18 + (2/3)

Now, we need to find a common denominator for the fractions on the right side:

(1/3)x = (-18 * 3 + 2)/3

Simplifying the expression on the right side:

(1/3)x = (-54 + 2)/3

(1/3)x = -52/3

To eliminate the fraction, we can multiply both sides of the equation by 3:

3 * (1/3)x = 3 * (-52/3)

This simplifies to:

x = -52

Answer:

Coefficients:  6, 1, -2

Variables:  x, y

Constants:   32

Step-by-step explanation:

Before identifying the coefficients, variables, and constants of the given algebraic expression, 6x + y + 32 - 2x, we must first define the required vocabulary terms.

6x + y + 32 - 2x

In the folllowing terms:

Coefficients:  6, 1, -2

Variables: x, y

Constants: 32

The value of n in the global truncation error of the Runge-Kutta method can be determined by taking the number of steps required to reach a certain point.

As the local truncation error is of O(h3), it means that the error in each step is proportional to h3. Therefore, if we take n steps, the total error would be proportional to h3n. Since we are given that the global truncation error is of O(hn), we can conclude that n must be equal to 3.

Based on the information provided, the local truncation error in the Runge-Kutta method is given as O(h^3). The global truncation error is generally one order lower than the local truncation error. Therefore, in this case, the global truncation error in the Runge-Kutta method can be determined as O(h^2), where the value of n is 2.

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Answer:

0.67

Step-by-step explanation:

correct

Gambler A loses his comparative advantage the more coins that are tossed for both Gamblers. But Gambler A will always maintain the advantage and has quite a heavy advantage when a few coins are tossed.

What is the binomial distribution?

The discrete probability distribution of the number of successes in a series of n independent experiments is called the binomial distribution with parameters n and p in probability theory and statistics.

Let X be the count of successes in the first n flips by A, Y the indicator that the last flip by A is also a success, and Z the count of successes in n flips by B.

Because Y is independent of X and Z, then:

P(X + Y > Z) = P(Y = 1) P(X ≥ Z) + P(Y = 0) P(X > Z)

= 1/2(P(X = Z) + 2P(X > Z))

P(X > Z) = P(Z > X)

Gambler A loses his comparative advantage the more coins that are tossed for both Gamblers. But Gambler A will always maintain the advantage and has quite a heavy advantage when a few coins are tossed.

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there are 6 different kinds of 2-topping pizzas that are available. Answer: 6 kinds.

To find the number of different kinds of 2-topping pizzas that are available, we need to use the combination formula.

The number of ways to choose r items from a set of n items is given by the formula:

C(n,r) = n! / (r! * (n-r)!)

where n! means n factorial, which is the product of all positive integers up to n.

In this case, we want to choose 2 toppings from a set of 4 toppings, so we have:

C(4,2) = 4! / (2! * (4-2)!)

= 6

what is number?

A number is a mathematical object used to represent a quantity or value. Numbers can be classified into different types based on their properties and characteristics.

Some common types of numbers include:

Natural numbers: positive integers (1, 2, 3, ...)

Whole numbers: all positive integers and 0 (0, 1, 2, 3, ...)

Integers: positive and negative whole numbers (-3, -2, -1, 0, 1, 2, 3, ...)

Rational numbers: numbers that can be expressed as a ratio of two integers (such as 1/2, -3/4, 5, etc.)

Irrational numbers: numbers that cannot be expressed as a ratio of two integers and have a non-repeating decimal expansion (such as π, √2, etc.)

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Answer:

permutations: order matters

combinations: order doesn't matter

Answer:

A combination doesn't need to be in a specific order, unlike permutation which needs to go in order.

Step-by-step explanation:

The coffee will reach a temperature of 143 degrees Fahrenheit in approximately 18.67 minutes.

Newton's law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. Mathematically, we can write this as:

dT/dt = -k(T - T_s)

where dT/dt is the rate of change of temperature with respect to time, T is the temperature of the object, T_s is the temperature of the surroundings, and k is a constant of proportionality.

In this problem, we are given that the coffee has a temperature of 200 degrees Fahrenheit when freshly poured and cools to 188 degrees Fahrenheit after 2 minutes in a room at 80 degrees Fahrenheit. Let's call the time at which the coffee reaches a temperature of 143 degrees Fahrenheit as t.

Using Newton's law of cooling, we can set up an equation as:

(dT/dt) = -k(T - T_s)

Integrating both sides, we get:

ln(T - T_s) = -kt + C

where C is a constant of integration.

Using the initial conditions, we can solve for C:

ln(200 - 80) = -k(0) + C

C = ln(120)

So the equation becomes:

ln(T - 80) = -kt + ln(120)

Now we can use the fact that the temperature of the coffee has cooled to 188 degrees Fahrenheit after 2 minutes to find the value of k:

ln(188 - 80) = -k(2) + ln(120)

k = (ln(120) - ln(108)) / 2

k ≈ 0.0776

Now we can solve for the time t when the coffee reaches a temperature of 143 degrees Fahrenheit:

ln(143 - 80) = -0.0776t + ln(120)

t ≈ 18.67 minutes

Therefore, the coffee will reach a temperature of 143 degrees Fahrenheit in approximately 18.67 minutes.

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Answer: The answer is of course, 3.

Step-by-step explanation:

Answer:

Your correct answer would be option 2.   y = IxI -1  Please vote Brainliest!

Step-by-step explanation:

Answer:

X is 6 or -6

Step-by-step explanation:

In absolute value it makes everything positive

Answer:

x = 6

Step-by-step explanation:

I hope this helps!

The simplified expression for the complex expression is -56-52i

What is a complex expression?

A complex expression is an expression of the form a+ ib where a , b are constant and i is the imaginary number known as iota. In a + ib a is the real part and b is the imaginary part.

We are given two complex number as (8-8i) and (8+4i)

We have to simplify the given expression

As Complex number are added multiplied subtracted same as real number we can simplify the expression easily

2(8-8i)-9(8+4i)

We multiply the constants inside the bracket

16-16i-72-36i

Separating the terms and simplifying them we get

-56-52i

Hence the simplified expression is -56-52i

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The largest subset S on which the vector-valued function F is continuous is the interval (-3, ∞).

To find the largest subset S on which the given vector-valued function is continuous, we need to examine the individual component functions and identify any restrictions or conditions that could cause discontinuities.

Component 1: f1(t) = t/(9 + t^4)

This function is a rational function. It will be continuous for all real values of t except where the denominator equals zero: 9 + t^4 = 0. However, there are no real values of t that satisfy this equation, so f1(t) is continuous for all real values of t.

Component 2: f2(t) = 7t^2 - 7t + 1

This function is a polynomial function, and polynomial functions are continuous for all real values of t. Therefore, f2(t) is continuous for all real values of t.

Component 3: f3(t) = ln(t + 3)

The natural logarithm function is defined for the positive values of its argument. Therefore, t + 3 > 0, which implies t > -3. Hence, the function f3(t) is continuous for t > -3.

Taking all these considerations into account.The function is continuous for all values of t greater than -3.

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Answer:

LCM is 36

Step-by-step explanation:

9 x 4 = 36

12 x 3 = 36

Answer:

35.09°

Step-by-step explanation:

You can use cos theta to find the value of x.

Let us solve now.

They have already given the lengths of the hypotenuse and adjacent.

Hypotenuse = 33

Adjacent = 27

Let us solve now.

Cos x = Adjacent ÷ Hypotenuse

Cos x = 27 ÷ 33

Cos x = 0.8181

x = Cos⁻¹ 0.8181

x = 35.09°

Hope this helps you :-)

Let me know if you have any other questions :)

Answer:

-2 1/3

Step-by-step explanation:

To find the opposite of a number, you just multiply it by -1 and that's the opposite. Let me know if that makes sense.

In other words, you just add a negative if it's a positive number and remove the negative if it's a negative number

Yes, it is possible to factor the expression x^2+4x-4. The expression can be factored as (x+2)(x-2).

To factor this expression, you can use the factoring method known as grouping. First, you can rewrite the expression as x^2 + 4x - 4 = (x^2 - 4) + 4x. Next, you can group the first two terms (x^2 - 4) and the last two terms (4x) into pairs of binomials, like this: (x^2 - 4) + 4x = (x+2)(x-2).

Alternatively, you could also use the difference of squares factorization to factor the expression. In this method, you would rewrite the expression as x^2 + 4x - 4 = (x^2 - 4x) + 4x = x(x-4) + 4(x-1). This can be further simplified as (x-4)(x+1).

Both of these methods are valid ways to factor the expression x^2 + 4x - 4, and either one would give you the correct answer of (x+2)(x-2).

Answer:

The slope is -1

Step-by-step explanation:

We need two points on the line to find the slope

(0,4) and (4,0)

The slope is

m = (y2-y1)/(x2-x1)

   = (0-4)/(4-0)

  = -4/4

  = -1

The exact value of tan[π/4 − cos⁻¹(7/25)] is 31/7.

We have to find the exact value of the expression.

tan[π/4 − cos⁻¹(7/25)]tan[π/4 − cos⁻¹(7/25)]

is an angle where we will use the following identities given below in order to solve the question.

Inverse cosine identity

cos⁻¹ x = π/2 − sin⁻¹ x , -1 ≤ x ≤ 1

Therefore, cos⁻¹(7/25) = π/2 − sin⁻¹(24/25)

Next, we will solve π/4 − [π/2 − sin⁻¹(24/25)]π/4 − [π/2 − sin⁻¹(24/25)]

can be written as (π/4) + sin⁻¹(24/25).

Thus,tan[π/4 − cos⁻¹(7/25)] = tan[(π/4) + sin⁻¹(24/25)]

Now, we have to use the trigonometric identity

tan(x + y) = (tan x + tan y) / (1 - tan x tan y)

where, x = π/4 and y = sin⁻¹(24/25).

We know the values of tan(π/4) and tan(sin⁻¹(24/25)).tan(sin⁻¹(24/25)) = 24/7

Therefore,tan[π/4 − cos⁻¹(7/25)] = tan[(π/4) + sin⁻¹(24/25)]

(tan (π/4) + tan (sin⁻¹(24/25))) / [1 - tan (π/4)tan (sin⁻¹(24/25))]

Using the values,

tan[π/4 − cos⁻¹(7/25)] = (1 + (24/7)) / [1 - (1 * 24/7)]

tan[π/4 − cos⁻¹(7/25)] = 31/7

Hence, the exact value of tan[π/4 − cos⁻¹(7/25)] is 31/7.

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Answer:

  9

Step-by-step explanation:

The 79-digit number of interest can be formulated as a sum of shorter numbers whose remainders can be computed.

The expanded form of the number can be written as ...

  a_44 = 01×10^78 +23×10^76 +45×10^74 +67×10^72 +89×10^70 +...

  +10×10^68 +11×10^66 +... +43×10^2 +44×10^0

Each number except the last is multiplied by a power of 10. Powers of 10 modulo 45 are ...

  10 mod 45 = 10

  100 mod 45 = 10

This lets us conclude that any positive power of 10 mod 45 is 10.

In short, all of the multiplication by powers of 10 can be collapsed to a single multiplication by 10. Hence, the mod 45 value of a_44 will be ...

  a_44 mod 45 = (((01 +23 +45 +67 +89) +10 +11 +12 +... +43)×10 +44) mod 45

  = (((01 +23 +45 +67 +89) mod 45 + sum(10 .. 43) mod 45)×10 +44) mod 45

  = (((225 mod 45) +(901 mod 45))×10 +44) mod 45

  = ((0 +1)×10 +44) mod 45

  a_44 mod 45 = 54 mod 45 = 9

__

Additional comment

The result can be confirmed by a suitable calculator.

__

The sum of the 34 numbers from 10 to 43 is the product of their average value (10+43)/2 = 26.5 and their number, 34. (26.5×34) = 901.

Answer: 9

Step-by-step explanation:

we can manually do this by dividing 1234567891011121314151617181920212223242526272829303132333435363738394041424344

on the calculator. I saw multiple hate comments on the answer above, and I wanted to test it manually - it still is 9.

Answer:

x = 80

Step-by-step explanation:

Tangent Chord Angle  =1/2 Intercepted Arc

40 = 1/2 x

80 = x

There are 5 full square and 6 triangles that are half squares.

      5 + (6)(1/2) = 5 + 3 = 8

You could also break this into smaller shapes (like a big triangle on the top and a smaller triangle and rectangle on the bottom and use area formulas to calculate the area.  But counting works well in this example.

Answer: 8

Step-by-step explanation:

First you split up this shape into two different shapes, a triangle and trapezoid

Put a line through the coordinates (-2,2) to  (-1,2); the top is a triangle and the other is a trapezoid

Area of the trapezoid is A = .5x (base1 + base2) x height

base1 of the trapezoid goes from -4 to -1 which is 3

base2 goes from -2 to -1 which is 1

height is 0 to 2 whcich is 2

A = .5 x (3+1) x 2 = 4

Now area of a triangle is A = .5 x base x height

the base goes from -2 to 2 which is 4

the height goes from 2 to 4 which is 2

A = .5 x (4) x (2) = 4

Area of the Trapezoid + Area of the Triangle = Total Area

4 + 4 = 8

mhorn seems to be correct

Step-by-step explanation:

yshahaa