find a value of c so that p(z ≥ c) = 0.55.

The general solution of the differential equation is: y = c1 e^(3t) cos(t) + c2 e^(3t) sin(t), where c1 and c2 are arbitrary constants determined by the initial or boundary conditions.

The given differential equation is a second-order homogeneous linear differential equation with constant coefficients. Its characteristic equation is obtained by assuming a solution of the form y = e^(rt), where r is a constant. Substituting this into the differential equation, we get:

r^2 e^(rt) - 6r e^(rt) + 10 e^(rt) = 0

Dividing both sides by e^(rt) (which is non-zero), we get:

r^2 - 6r + 10 = 0

Solving for r using the quadratic formula, we get:

r = (6 ± sqrt(6^2 - 4(1)(10))) / 2

r = 3 ± i

Therefore, the general solution of the differential equation is:

y = c1 e^(3t) cos(t) + c2 e^(3t) sin(t)

where c1 and c2 are arbitrary constants determined by the initial or boundary conditions.

This solution consists of a linear combination of two functions: the exponential function e^(3t) and the periodic function cos(t) or sin(t), which oscillates between -1 and 1. The exponential function represents the growth or decay of the system, while the periodic function represents the oscillations or vibrations of the system. The constants c1 and c2 determine the amplitude and phase of the oscillations, respectively. The behavior of the system depends on the signs and magnitudes of the real and imaginary parts of the roots, which determine whether the solutions are overdamped, critically damped, or underdamped.

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The interest rate being earned on the college education fund set aside by your parents 14 years ago is 8.51 percent. This rate of interest has resulted in the value of the fund increasing from $7,500 to $26,180 over the period.

The calculation of the interest rate is based on the concept of compound interest, where the interest earned is added to the initial principal amount, and subsequent interest is calculated on the new balance. Using the formula for compound interest, the rate can be determined as follows:

$26,180 = $7,500(1 + r/100)^14

Solving for the interest rate, r, gives a value of 8.51 percent. This means that the fund has been growing at a rate of 8.51 percent per year for the past 14 years. It is worth noting that the actual rate of return may vary due to factors such as fees, taxes, and market fluctuations, but the calculated rate provides a good estimate of the average growth rate of the fund.

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The measure of arc BED is:

180 degrees

The arc of a circle is the segment or part of the circumference of a circle. The measure of an arc is the angle at which the arc is subtended.

You will notice that AD is the diameter of the circle.

Since the measure of arc CD is 90°.

Thus, the measure of arc AB is:

90 - 52 =  38°

Arc ED = arc AB (vertically opposite angles are equal)

Therefore, the measure of Arc BED will be:

Arc BED = Arc BC + arc CD + arc ED

Arc BED = 52° +  90° +  38°

Arc BED = 180°

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The correct expression to solve for the amount of time it takes to drain the hot tub is: A) (9t - 9)(5t - 7)

In Mathematics and Geometry, the general form of a quadratic function can be modeled and represented by using the following quadratic equation;

y = ax² + bx + c

Where:

By using the sum-product pattern, we have the following:

45t² − 108t + 63

45t² − 63t - 45t + 63

By writing the common factor from the two pairs, we have the following:

(45t² − 63t) - (45t + 63)

9t(5t - 7) - 9(5t - 7)

By rewriting in factored form, we have the following:

(9t - 9)(5t - 7)

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Answer: Range = 22 IQR = 13

Step-by-step explanation:

To find the range of a data set, subtract the largest value of the data set from the smallest value. 45 is the largest value and 23 is the smallest, and 45-23 is 22, so the range is 22. To find the IQR, you must subtract the upper quartile and the lower quartile. We can find this after finding the mean by re-arranging all the values in order from lowest to greatest, and finding the middle value. Once we do so, we get 33.5. Then, we find the median of all the numbers below and after the median to get the lower and upper quartiles. Then, we subtract them to get the IQR 13.

Answer:

y = (3/2)x + 1

Step-by-step explanation:

straight line equation format is y -y1 = m(x - x1)

where y1 is y-coordinate of a point, x1 is x-coordinate of the same point, m is the slope (gradient).

slope of y - 4 = -2/3 (x - 6) is -2/3, since that's the number before x.

this means that slope of perpendicular line = -(1/ (-2/3)) = 3/2.

it passes through (-2, -2)

equation is y - -2 = 3/2 (x - - 2)

y + 2 = 3/2 (x +2) = (3/2) x + 3

subtract 2 from both sides

y = (3/2)x + 1

Answer:$7.50

Step-by-step explanation:The original cost of the sandals is $30 and the discount if is 75%.That means only 25% of the original cost of the sandals remain so we do 30x.25=$7.50

The radius and diameter of each measure to their approximation is as follows;

7) 14 km   8) 5 yards   9) 23 inches   10) 20 feet  11) 17.5 mm  12. 22.5 cm

13) 95feet²  14) 92.2 mm 15) 119.6 m  16)227 inches²

For each problem, we use the formulas;

Area of a circle: A = πr²

Circumference of a circle: C = 2πr or C = πd

a) Find the radius of a circle with an area of 615.75 square kilometers.

A = πr²

∴ r = √(A/π)  

√(615.75/π)

= 14 km

b) Find the diameter of a circle with a circumference of 15.71 yards.

C = πd

∴ d = C/π

15.71/π  = 5 yards

c) Find the diameter of a circle with an area of 415.48 square inches.

A = πr²

∴ r = √(A/π)

√(415.48/π) = 11.5 inches,

∴ d = 2r = 23 inches

d) Find the radius of a circle with a circumference of 125.66 feet.

C = 2πr

∴ r = C/(2π)

125.66/(2π) = 20 feet

e) Find the diameter of a circle with an area of 240.25 square millimeters.

sqrt(240.25/π) = 8.75 mm,

∴ d = 2r = 17.5 mm

f) Find the radius of a circle with a circumference of 45π centimeters.

45π/(2π) = 22.5 cm

g) Find the area of a circle with a circumference of 11π feet.

11π/(2π) = 5.5 feet,

A = πr² = π×(5.5)² = 95 feet²

h) Find the circumference of a circle with an area of 676 square millimeters.

A = πr²

∴ r = √A/π) =√(676/π) = 14.67 mm,

∴ C = 2πr = 2π×14.67 = 92.2 mm

i. Find the circumference of a circle with an area of 1,134.11 square meters.

√(1,134.11/π) = 19.03 m,

C = 2πr = 2π×19.03 = 119.6 m

j. Find the area of a circle with a circumference of 53.41 inches.

C = 2πr

∴ r = C/(2π) = 53.41/(2π) = 8.5 inches,

∴ A = πr² = π×(8.5)² = 227inches²

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the rate of change of the radius at the instant when the radius is 11 feet and the volume is 26 cubic feet is -10 feet per second.

What is Volume?

Volume is the amount of three-dimensional space enclosed by a closed surface, such as the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using an SI-derived unit, the cubic meter

To find the rate of change of the radius at the instant when the radius of the cylinder is 11 feet and the volume is 26 cubic feet, we can use the related rates of change between the height, radius, and volume of the cylinder.

Let's denote the height of the cylinder as h, the radius as r, and the volume as V. We are given that the height is decreasing at a constant rate of 2 feet per second, and the volume is decreasing at a rate of 264 cubic feet per second.

We know the formula for the volume of a cylinder is V = πr^2h.

Differentiating both sides of the equation with respect to time (t), we have:

dV/dt = d(πr^2h)/dt

Using the product rule, we can rewrite this as:

dV/dt = π(2rh(dr/dt) + r^2(dh/dt))

Since we are given the values of dV/dt (rate of change of volume) and dh/dt (rate of change of height), and we want to find the rate of change of the radius, we can substitute these values into the equation along with the known values of r (radius) and V (volume) at the given instant.

dV/dt = -264 (volume is decreasing at a rate of 264 cubic feet per second)

dh/dt = -2 (height is decreasing at a rate of 2 feet per second)

r = 11 (radius is 11 feet)

V = 26 (volume is 26 cubic feet)

Plugging in these values, we have:

-264 = π(2(11)(dr/dt) + 11^2(-2))

Simplifying further:

-264 = π(22(dr/dt) - 22)

Dividing both sides by π(22), we get:

-12 = dr/dt - 2

Adding 2 to both sides, we have:

dr/dt = -12 + 2 = -10

Therefore, the rate of change of the radius at the instant when the radius is 11 feet and the volume is 26 cubic feet is -10 feet per second. Note that the negative sign indicates that the radius is decreasing.

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[tex]3(x - 2) - 2x < 18\\3x-6-2x < 18\\x < 24[/tex]

The perimeters and areas of each figures are

Given that all curves are semicircles or quadrants., we have

Figure 1

The area is calculated as

Area = 1.6 * 2.8

Area = 4.48 m²

The perimeter is calculated as

Perimeter = 2 * (1.6 + 2.8) + 3.14 * 1.6/2

Perimeter = 11.312 meters

Figure 2

The area is calculated as

Area = 50 * 68

Area = 3400 cm²

The perimeter is calculated as

Perimeter = 2 * (50 + 68) + 3.14 * 50

Perimeter = 393 centimeters

Figure 3

The area is calculated as

Area = 3.14 * 5² + 2 * 3.14 * (5/2)² * 1/2

Area = 98.125 m²

The perimeter is calculated as

Perimeter = 2 * 3.14 * 5 + 2 * 1/2 * 3.14 * 5/2

Perimeter = 39.25 meters

Figure 4

The area is calculated as

Area = 3.14 * (4.6/2)²/2 + 3.14 * (2.3)²/4

Area = 12.46 cm²

The perimeter is calculated as

Perimeter = 3.14 * (4.6/2) + 3.14 * (2.3)/4

Perimeter = 9.03 centimeters

Figure 5

The area is calculated as

Area = 3.14 * (15)²/2 + 2 * 3.14 * (5)²/2 - 3.14 * (5)²/2

Area = 392.5 cm²

The perimeter is calculated as

Perimeter = 3.14 * (15) + 2 * 3.14 * (5)/2 - 3.14 * (5)/2

Perimeter = 54.95 centimeters

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To ensure that you have at least two cards with the same denomination, you would need to draw five cards.

This problem is a variation of the pigeonhole principle, also known as the birthday paradox. There are 13 denominations in a standard deck of cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King), and drawing five cards creates five pigeonholes. Since there are more pigeonholes than there are denominations, it is guaranteed that at least two of the cards will have the same denomination. To see this, note that drawing six cards will also guarantee two cards with the same denomination, since there would be six pigeonholes and only 13 denominations.

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We can estimate that Cho will sell around 100 servings of kakigori on a day when the high temperature is 29° Celsius.

Based on the line of best fit for the data, we can estimate the number of servings of kakigori Cho will sell on a day when the high temperature is 29° Celsius.

From the scatter plot, we can see that the line of best fit is increasing as the temperature increases.

By estimating the value on the line of best fit for the temperature of 29° Celsius, we can approximate the number of servings sold. Based on the scatter plot, it appears that the number of servings sold is around 100 when the temperature is 29° Celsius.

Therefore, we can estimate that Cho will sell around 100 servings of kakigori on a day when the high temperature is 29° Celsius.

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For the given function the answer is a + b = 3/2 + 9/2 = 12/2 = 6.

A function is an association between inputs in which each input has a unique link to one or more outputs.

To find the values of a and b, we can substitute the expressions for f(x) and g(x) into the equation g(f(x)) = 3x + 4 and solve for a and b.

We have:

g(f(x)) = 3x + 4

Substituting the expressions for f(x) and g(x), we get:

g(ax + b) = 3x + 4

Since g(x) = 2x - 5, we can replace g(ax + b) with 2(ax + b) - 5:

2(ax + b) - 5 = 3x + 4

Expanding and simplifying, we have:

2ax + 2b - 5 = 3x + 4

Rearranging terms, we get:

(2a - 3)x + (2b - 9) = 0

For this equation to hold for all values of x, the coefficient of x must be zero, and the constant terms must also be zero.

Therefore, we have the following system of equations:

2a - 3 = 0   (coefficient of x)

2b - 9 = 0   (constant term)

Solving these equations simultaneously, we find:

2a = 3   -->   a = 3/2

2b = 9   -->   b = 9/2

Therefore, a + b = 3/2 + 9/2 = 12/2 = 6.

Hence, a + b = 6.

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Answer: 14/5

Step-by-step explanation:

2.8 = 28/10

this can be simplified to 14/5

Answer:

14/5

Step-by-step explanation:

The value of a = 1/3 and b = 2.

We can expand the given expression using the binomial theorem:

Binomial theorem statement that for any positive integer n the nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form in the sequence of terms, the index r takes on the successive values 0, 1, 2,…, n.

Equation.

(2-x)(1+ax)⁶ = (2-x) × [1 + 6a x + 15a² x² + ...]

Multiplying this out gives:

(2-x)(1+ax)⁶ = 2 + (6a-2x) x + (15a² - 12ax + x²) x² + ...

Comparing the coefficient of x and x² with the given expansion we get:

6a - 2 = 0 (coefficient of x)

15a² - 12a + 1 = b (coefficient of x²)

From the first equation, we get:

6a = 2

a = 1/3

Substituting this value of a into the second equation, we get:

15(1/3)² - 12(1/3) + 1 = b

5 - 4 + 1 = b

b = 2

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Answer:

B

Step-by-step explanation:

The positive 9 shows that it's moving up 9 units, the negative in front of the square root reflects the equation over the x-axis, and the positive 3 shows that it has been moved to the left 3 units.

Outside of equation: positive numbers translate up, negative translate down. (ex: [tex]x^{2}+3, x^{2}-3[/tex]. The three would move the equation up or down.)

Inside of equation: positive numbers translate left, negative translate right. (ex: [tex](x-3)^{2},(x+3)^{2}[/tex]. The 3 would move left or right.)

The perimeter of the tree house floor is  27.62 meters and area of the tree house floor is 30 square meters.

To find the perimeter of the tree house floor, we need to add up the lengths of all four sides of the rectangular floor.

We can use the distance formula to calculate the length of each side:

Length of side AB =√(6-0)² + (4-4)²) = √(36) = 6 meters

Length of side BC =√(6-0)² + (-1-4)²) = √61 = 7.81 meters

Length of side CD = √(0-6)² + (-1-4)² = √61= 7.81 meters

Length of side DA = √(0-6)²+ (4-4)²) =√36= 6 meters

Therefore, the perimeter of the tree house floor is:

Perimeter = AB + BC + CD + DA

Perimeter = 6 + 7.81 + 7.81 + 6

Perimeter = 27.62 meters

To find the area of the tree house floor, we can use the formula for the area of a rectangle:

Area = base x height

The base of the rectangle is the length of side AB, which is 6 meters.

The height of the rectangle is the distance between points B and C, which is 4-(-1) = 5 meters. Therefore, the area of the tree house floor is:

Area = base x height

Area = 6 x 5

Area = 30 square meters

Hence, the area of the tree house floor is 30 square meters.

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Answer:

The function f(x) = cos^2x + sin^2x is not a straight line, but rather a constant function that always evaluates to 1 for any value of x.To see why this is the case, recall the trigonometric identity that states that the square of the cosine of an angle added to the square of the sine of the same angle equals 1:cos^2x + sin^2x = 1Since this identity holds for all values of x, it follows that f(x) = 1 for all x. Therefore, the graph of this function is a horizontal line at y = 1, not a straight line.

Step-by-step explanation:

The relationship described, where the number of ants in a colony doubles every month, can be modeled by an exponential function.

An exponential function is of the form y = ab^x, where 'a' is the initial value or starting point, 'b' is the growth factor, and 'x' represents the time or number of months in this case.

In the given scenario, the initial number of ants is multiplied by a growth factor of 2 every month. This corresponds to the base 'b' of the exponential function being equal to 2. The exponent 'x' represents the number of months elapsed.

Therefore, the exponential function y = 2^x best models the relationship between the number of ants in the colony and the time, as it reflects the doubling pattern observed in the colony's population over time.

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1. It will cost Dave $646.62 to refill his truck's fuel tank.

2. The cost of concrete for Greg's driveway will be $9,720.

3. Total cost of the topsoil needed from suppliers is $726.90 from Holland Growers and $919.55 from Night Growers.

A refill refers to filling something again with more of same thing such as fuel, water etc.

We need to convert the fuel capacity of Dave's truck from gallons to liters.

1 US gallon = 3.78541 liters

So, 180 US gallons will give us:

= 180 * 3.78541 liters

= 681.37 liters

We will calculate total cost of refilling Dave's truck which is:

= 681.37 liters * 0.949 dollars/liter

= $646.62013

= $646.62.

2. To get volume of concrete needed, we will convert thickness of the driveway from inches to meters. 4 inches is equal to 0.1 meters.

The volume of the driveway is:

V = l x w x h

V = 20m x 36m x 0.1m

V = 72 cubic meters

The cost of the concrete for the driveway is:

= Volume x Price per cubic meter

= 72 cubic meters x $135/cubic meter

= $9,720.

3. To get total cost of the topsoil, we have to calculate volume of soil required for each planter box:

Volume = Length x Width x Depth

Volume = 44" x 24" x 18"

Volume = (44/36) yards x (24/36) yards x (18/36) yards

Volume = 1.22 yards³

Since she needs to build 35 planter boxes, the total volume of soil required is:

= 1.22 yards³ x 35

= 42.7 yards³

The cost from Holland Growers will be:

= Total Volume x $17/yd

= 42.7 yards³ x $17/yd

= $726.90

The cost from Night Growers will be:

= Total Volume x $21.50/yd²

= 42.7 yards³ x $21.50/yd²

= $919.55.

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Answer:

54

Step-by-step explanation:

686779496080807

[tex]\sf D = 1260 \ m[120 \ o] \ CW \ from +y-axis.[/tex]

[tex]\sf Y = 1260\times Cos \ (120) = -630 = 630 \ m. \ South.[/tex]

Therefore, 630 m far south is he from his starting point​.

To create the greatest 5-digit number using the digits 4, 2, and 0, we need to place the highest value digit in the leftmost position. Therefore, the greatest 5-digit number that can be created using the digits 4, 2, and 0 is 42,000.  

By arranging the digits in this order, you ensure that you have the largest possible value while still using only the given digits 4, 2, and 0. In this case, the digit 4 is the highest value digit. So we place it in the ten-thousands place. This gives us 40,000. Now we need to fill in the remaining four digits. Since we want to create the greatest possible number, we need to use the next highest digit, which is 2, in the thousands place. This gives us 42,000. We can then use the remaining digit, which is 0, in the hundreds, tens, and ones places. to create the greatest 5-digit number, we need to start with the highest value digit and place it in the leftmost position. Then, we fill in the remaining digits with the next highest value digits in descending order. This ensures that we create the greatest possible number.

The greatest 5-digit number using the digits 4, 2, and 0, you can repeat the digits as needed. Place the largest digit (4) in the leftmost position to maximize the value, and then fill in the remaining positions with the next largest digit (2), followed by the smallest digit (0).

Therefore, the greatest 5-digit number that can be created using the digits 4, 2, and 0 is 42,000.

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Answer:

a is 90

b is 90

Step-by-step explanation:

I'm not 100 percent sure about A but I think it is right

Answer:

a=5 b=7.07

Step-by-step explanation:

a(sin45)=5sin45

a=5sin45/sin45

a=5

bsin45=5sin90/45

b=7.07

the above numbers (Saturday or Sunday) the true probability is closer to. Inverse correlation implies that as the probability of shopping on Saturday increases, the probability of shopping on Sunday decreases, and vice versa.

If the probability of shopping on Saturday is inversely correlated with the probability of shopping on Sunday, we can infer that as the likelihood of shopping on one day increases, the likelihood of shopping on the other day decreases.

Without specific numbers or additional information, it is impossible to determine which probability is closer to the true probability.

Let's consider two scenarios to illustrate this point.

In Scenario A, the probability of shopping on Saturday is high (e.g., 80%), suggesting a low probability of shopping on Sunday (e.g., 20%).

In Scenario B, the probability of shopping on Saturday is low (e.g., 20%), indicating a high probability of shopping on Sunday (e.g., 80%).

Both scenarios exhibit an inverse correlation between the two probabilities.

In Scenario A, the true probability is closer to 80% for shopping on Saturday, while in Scenario B, the true probability is closer to 80% for shopping on Sunday.

Thus, without specific values or additional information, it is not possible to determine which probability is closer to the true probability.

The inverse correlation only tells us about the relationship between the two probabilities, not their actual values.

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An angle in standard position measures π/2 radians, and p(0, 1) is on the terminal side of the angle, the value of the cosine of this angle is 1.

So, we have:

cos θ = adjacent/hypotenuse

cos θ = x/1

cos θ = x

Now, the given angle is π/2 radians, and the point P (0, 1) lies on the terminal side of the angle. From the point P (0, 1), we can move left towards the origin to make a right-angled triangle.

Since the angle measures π/2, one leg will be on the x-axis (the horizontal leg) and the other leg will be on the y-axis (the vertical leg). Let's construct a right triangle using the point P (0, 1) and the origin (0,0) as two of the vertices: So, the opposite side is equal to 0, and the adjacent side is equal to 1.

Thus, the value of the cosine of this angle is cos θ = adjacent/hypotenuse = 1/1 = 1.

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Answer:

0

Step-by-step explanation:

Answer:

not possible

Step-by-step explanation:

error

Answer:

a + c = 90° because BC is a tangent to circle A, meaning that angle B is a right angle.

Answer:

a ) -1

Step-by-step explanation:

[tex]\frac{1}{64} = 4^{2s-1} . 16^{2s+2}\\ 4^{-3} = 4^{2s-1+4s+4}\\ 4^{-3} = 4^{6s+3}\\ 6s + 3 = -3\\6s = -6\\s = -1[/tex]