A solid has as its base the region bounded by the curves y = –2x^2 + 2 and y =-x^2+ 1. Find the volume of the solid if every cross section of a plane perpendicular to the x-axis is a trapezoid with lower base in the xy-plane, upper base equal to 1/2 the length of the lower base, and height equal to 2 times the length of the lower base.

The water rises 10 inches each hour. The water is 4 inches deep is the water after four hours. It will take 8 hours for the water to reach a depth of 80 inches.

A pipe cracked overnight and started to leak into a basement.

The graph represents the depth (d) of the water over time (t).

As per the given graph, the required solution would be as:

Take two points (1, 10) and (5, 50) in the given graph

The linear function will be :

d - 10 = (50-10)/(5-1)(t - 1)

d - 10 = (40/4)(t - 1)

d = 10(t - 1) + 10

d = 10t - 10 + 10

d = 10t

Here slope of the given function is 10, which means 10 inches the water rise each hour.

The value of time (t) = 4 hr corresponds to the depth of water (d) is

d = 10(4) = 40

So, the water is 40 inches deep is the water after four hours

The value of depth of water (d) = 80 in corresponds to the time (t) is

80 = 10t

t = 80/10 = 8

So, It will take 8 hours for the water to reach a depth of 80 inches

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The perimeter and the area of rectangle EFGH are 24 cm and 35 square cm

From the question, we have the following parameters that can be used in our computation:

Side length, AB = 5 cm

Extension AE=AH=FC=GC = 2cm

Using the above as a guide, we have the following:

Dimension of the rectangle = 5 cm By (5 + 2) cm

So, we have

Dimension of the rectangle = 5 cm By 7 cm

The perimeter is then calculated as

P = 2 * (5 + 7) = 24 cm

The area is calculated as

A = 5 * 7 = 35 square cm

Hence, the perimeter and the area of rectangle EFGH are 24 cm and 35 square cm

See attachment for figure

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These two events are independent because the outcome of both doesn't affect each other.

An independent event is the type of event that occurs when the outcome of one event does not affect the outcome of the second event it is compared with.

A dependent event is the type of event that it's outcome affects the outcome of the second event it is compared with.

That Abigail chose an entree in a town different from Zeke doesn't affect her ability to attend the wedding.

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The 10th percentile is 164 milliseconds, and the 75th percentile is 261 milliseconds.

To find the percentiles for the given data, we first need to arrange the data in ascending order:

152, 164, 175, 193, 212, 217, 222, 235, 250, 257, 261, 273, 296, 311

There are 15 data points, so the 10th percentile can be calculated as follows:

10th percentile = (10/100) * (15 - 1) + 1 = 1.4

The 10th percentile corresponds to the 2nd data point in the ordered list:

10th percentile = 164

Similarly, the 75th percentile can be calculated as follows:

75th percentile = (75/100) * (15 - 1) + 1 = 11.4

The 75th percentile corresponds to the 11th data point in the ordered list:

75th percentile = 261

Therefore, the 10th percentile is 164 milliseconds, and the 75th percentile is 261 milliseconds.

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The perimeter of the plot land is 78.49 units and the area of the plot land without the building and parking lot is 210 square units

From the graph, the vertices of the land are

A = (0, 20), B = (25, 19), C = (20, 2) and D = (5, 0)

Calculate the distance between adjacent vertices

So, we have

AB = √[(0 - 25)² + (20 - 19)²] = 25.02

BC = √[(20 - 25)² + (2 - 19)²] = 17.72

DC = √[(20 - 5)² + (2 - 0)²] = 15.13

AD = √[(0 - 5)² + (20 - 0)²] = 20.62

The perimeter of the plot land is

P = 25.02 + 17.72 + 15.13 + 20.62

P = 78.49 units

This is calculated as

Area = Plot land - Building - Parking lot

Using the vertices, we have

Area = 1/2 * |0 * 19 + 25 * 2 + 20 * 0 + 5 * 20 - (20 * 25 + 19 * 20 + 2 * 5 + 0 * 0)| - [1/2 *(15 - 7 + 17 - 7) * (15 - 10)] - [(18 - 10) * (10 - 5)]

This gives

Area = 295 -  45 - 40

So, we have

Area = 210

Hence, the area is 210 square units

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The perimeter of the figure is 24.50 feet feet, and the area of the figure is 45.68 square feet

From the question, we have the following parameters that can be used in our computation:

Radius, r = 4 feet (see attachment)

The perimeter of the figure is

Perimeter = Circumference of 3/4 circle + Perimeter of triangle - 2r

So, we have

Perimeter  = 3/4 * 2 * 3.14 * 4 + 4 + 4 + 4√2 - 2 * 4

Evalate

Perimeter  = 24.50 feet

The area of the figure is:

Area = Area of 3/4 circle + Area of triangle

So, we have

Area = 3/4 * 3.14 * 4² + 1/2 * 4 * 4

Evaluate

Area = 45.68

Hence, the area is 45.68 square feet

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The true statement is that f(x) and g(x) have the same y-intercept. Option C.

Both f(x) and g(x) have the same y-intercept. We can see this by setting x = 0 for both functions.

         f(0) = -15(0)^2 + 32 = 32

       g(0) = -17(0)^2 + 5(0) + 32 = 32

The other statements are not necessarily true. f(x) and g(x) do not necessarily have the same zeros, as the coefficients and constants in the functions are different.

The maximum value of f(x) and g(x) would depend on the vertex of each function, which may or may not be the same.

Finally, neither f(x) nor g(x) are odd functions, as they do not satisfy the property of f(-x) = -f(x) or g(-x) = -g(x) for all x.

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Answer:

250 students and 310 adults

Step-by-step explanation:

so if  students are charged 1.25 and adults are charged 2.25 and 560 people attended if you divided from it should equal

Austin's rate of motion is (1/70)π Radians per second.

To determine Austin's rate of motion in radians per second, we need to use the formula for angular velocity:

ω = Δθ / Δt

Where:

ω = angular velocity (in radians per second)

Δθ = change in angular displacement (in radians)

Δt = change in time (in seconds)

We know that Austin runs three-quarters of a circular track, which means he covers an arc length that is equal to three-quarters of the circumference of the circle. Let's call the radius of the circle "r". Then, the arc length covered by Austin is given by:

s = (3/4) * 2πr

s = (3/2)πr

We also know that it takes Austin 1 minute and 45 seconds to cover this distance. This is the same as 105 seconds (since 1 minute = 60 seconds).

So, Δt = 105 seconds

Now, we can calculate the change in angular displacement (Δθ). The total angle around a circle is 2π radians, so the angle covered by Austin is given by:

Δθ = (3/4) * 2π

Δθ = (3/2)π

Therefore, Austin's rate of motion (ω) in radians per second is:

ω = Δθ / Δt

ω = [(3/2)π] / 105

ω = (3/210)π

ω = (1/70)π radians per second

So, Austin's rate of motion is (1/70)π radians per second.

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true or false if two squares have equal side length then they are congruent

The area of the triangle given at the end of the answer is of:

B. 24 units².

The area of a rectangle of base b and height h is given by half the multiplication of dimensions, as follows:

A = 0.5bh.

Considering the image presented for this problem, the dimensions for the triangle are given as follows:

b = 12, h = 4.

Hence the area is given as follows:

A = 0.5 x 4 x 12

A = 24 units².

This means that option B is the correct option in the context of this problem.

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

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Answer:

0.17

Step-by-step explanation:

Probability for Event A: 1/3

Probability for Event B: 1/2

Total Probability: 1/6

Answer:

Step-by-step explanation:

Let's set David's score to D, and Aaron's score to A. If so, we get the equation that satisfies the constraints:

D = 3*A-5, D + A = 215

Set D in terms of A:

D = 215 - A

Input it into the equation:

215 - A = 3*A - 5 -> add 5 and A to both sides to set numbers to the left and variables to the right

215 + 5 = 3* A + A ->

220 = 4A -> divide by 4

220/4 = A ->

55 = A -> substitute A into D + A = 215

D + 55 = 215 -> subtract 55 from both sides

D = 215 - 55 = 160

So, out final answer is:

D = 160, A = 55

The value of permutation expression ₅₉P₂ is 3422.

A permutation is an arrangement of objects in a definite order.

To find the value of ₅₉P₂, we use the formula below

Formula:

Where:

Substitute these values into equation 1

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The total surface area of the triangular prism is is 1872 square meters

From the question, we have the following parameters that can be used in our computation:

The triangular prism

The surface area of the triangular prism from the net is calculated as

Surface area = sum of areas of individual shapes that make up the of the triangular prism

Using the above as a guide, we have the following:

Area = 1/2 * 12 * 16 * 2 + 35 * (20 + 16 + 12)

Evaluate

Area = 1872

Hence, the surface area is 1872 square meters

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Answer:

train is the mode of transportation that can be use

Answer:

The length = 65 The width = 80

Step-by-step explanation:

w = 15^2 = 290

225 =290

290-225 =80

w=80

80 - 15 = 65

l = 65

65+65+80+80=290

Answer:

Length: 65

Width: 80

Step-by-step explanation:

Lets turn it into an algebraic expression. x is the length.

length width

x           x+15

2x + 2(x+15) = 290

2x +2x +30 =290

4x = 260

x = 65

65+15 = 80

The %RE value for the Improved Euler (Heun) method is zero, which means that the method is exact. The %RE value for the Explicit Euler's method is 0.14%, which means that the method is a good approximation.

First rewrite the differential equation as follows:

[tex]\frac{dy}{dt } = (\frac{4}{t } - 6t^2)y[/tex]

Now integrate both sides of the equation:

[tex]\int\limits{\frac{dy}{dt}} \, = \int\limits\, (\frac{4}{t } - 6t^2)y dt[/tex]

This gives us:

[tex]y = 4ln(t) - 2t^3 + C[/tex]

where C = arbitrary constant.

Now, use the initial condition y(1) = 2 to find the value of C:

[tex]2 = 4ln(1) - 2(1)^3 + C[/tex]

This gives us C = 6.

Therefore, the solution of the initial-value problem is:

[tex]y = 4ln(t) - 2t^3 + 6[/tex]

Now, use the Explicit Euler's and Improved Euler (Heun) methods to compute y(1.3) numerically:

Explicit Euler's method:

First find the step size h:

h = (1.3 - 1) = 0.3

Now, use the Explicit Euler's method to compute y(1.3) as follows:

[tex]y(1.3) = y(1) + \frac{h}{2} * \frac{dy}{dt}[/tex]

[tex]y(1.3) = 2 + 0.3 * (4ln(1.3) - 2(1.3)^3 + 6)[/tex]

y(1.3) = 2.119

Improved Euler (Heun) method:

First find the step size h:

h = (1.3 - 1) = 0.3

Now, use the Improved Euler (Heun) method to compute y(1.3) as follows:

[tex]y(1.3) = y(1) + \frac{h}{2} * (\frac{dy}{dt} + \frac{dy}{dt'})[/tex]

where dy/dt' is the value of dy/dt evaluated at t = 1.3:

[tex]\frac{dy}{dt'} = 4ln(1.3) - 2(1.3)^3 + 6[/tex]

[tex]y(1.3) = 2 + \frac{0.3}{2} * (4ln(1.3) - 2(1.3)^3 + 6 + 4ln(1.3) - 2(1.3)^3 + 6)[/tex]

y(1.3) = 2.122

Now, calculate the %RE values for both methods:

Explicit Euler's method:

%RE = [tex]\frac{(y(1.3) - y_{true})}{y_{true}} * 100[/tex]

where y_true is the exact value of y(1.3) = 2.122:

%RE = (2.119 - 2.122)/2.122 × 100 = 0.14%

Improved Euler (Heun) method:

%RE = [tex]\frac{(y(1.3) - y_{true})}{y_{true}} * 100[/tex]

%RE = (2.122 - 2.122)/2.122 × 100 = 0%

The %RE value for the Improved Euler (Heun) method is zero, which means that the method is exact. The %RE value for the Explicit Euler's method is 0.14%, which means that the method is a good approximation.

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Answer:

length = 7 width = 2

Step-by-step explanation:

for the perimeter 7+7+2+2 is 18 and the area is 7×2 is 14 so the length is 7 and width is 2

After considering all the given data we conclude that the range is 45, the variance is 422.21, the standard deviation is 20.55

To evaluate the range of the given sample data, we subtract the smallest value from the largest value. For the given case, the largest value is 79 and the smallest value is 34. Therefore, the range is 79 - 34 = 45.
To evaluate the variance of the given sample data, we first need to find the mean. The mean is evaluated by adding up all of the values and dividing by the number of values. Including the given case, we have 10 values, so we add them up and divide by 10:
(51 + 63 + 36 + 43 + 34 + 62 + 73 + 39 + 53 + 79) / 10
= 51.3

Finally , we have to apply subtraction the mean from each value and square the result. Then we add up all of these squared differences and apply division by n-1 (here n is the number of values). This gives us the variance:
((51 - 51.3)² + (63 - 51.3)² + (36 - 51.3)² + (43 - 51.3)² + (34 - 51.3)² + (62 - 51.3)² + (73 - 51.3)² + (39 - 51.3)² + (53 - 51.3)² + (79 - 51.3)²) / (10-1) = 422.21
Lastly,  to evaluate the standard deviation, we take the square root of the variance:
√(422.21) = 20.55
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The circumference of the circle 48.4 inches.

The circumference of a circle is the perimeter of the circle. Therefore, let's find the circumference of the circle as follows:

length of arc = ∅ / 360 × 2πr

where

Therefore, let's find the radius

length of arc = 35 / 360 × 2 × 3.14 × r

4.6 = 219.8 / 360 r

4.6 = 0.61055555555 r

divide both sides by 0.61

r = 4.6 / 0.6

r = 7.7 inches

Therefore,

circumference = 2 × 3.14 × 7.7

circumference = 48.356

circumference = 48.4 inches

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Average is the total sum of the given values divided by the number of values.

The average low temperature for the week is -3°F.

We have,

It is the total sum of the given values divided by the number of values.

We have,

Temperatures for one week:

Day                   Temperature

Sunday          −5 °F ​

Monday         −8 ​°F ​

Tuesday         −6​°F ​

Wednesday     −8​°F ​

Thursday         −3 ​°F

Friday                2° F

Saturday           7° F

The average of the low temperature:

= [ -5 + (-8) + (-6) + (-8) + (-3) + 2 + 7 ] / 7

= (-5 - 8 - 6 - 8 - 3 + 2 + 7) / 7

= -30 + 9 / 7

= -21 / 7

= -3

Thus,

The average low temperature for the week is -3°F.

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complete question:

Christopher recorded the low temperature each day for 1 week and recorded his results in this table.

Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Low −5 °F ​ −8 ​°F ​ −6 ​°F ​ −8​°F ​ −3 ​°F 2° F 7° F

What is the average low temperature for the week?

Answer:

D.

Step-by-step explanation:

6a² = surface area for cube

6(5)² = 150

2[tex]\pi[/tex](r)(h)+2[tex]\pi[/tex]r²  = surface area cylinder

2[tex]\pi[/tex](2)(3)+2[tex]\pi[/tex](4) = 20pi

150 + 20pi = 212.8318531

The minimum speed at which the car was traveling is 52.9 miles per hour.

To find the minimum speed at which the car was traveling during the skid, we can use the formula for calculating skid speed:

Skid speed = (Chord length x Chord length x Drag factor) / (2 x Middle ordinate)

Plugging in the given values:

Chord length = 70 feet

Middle ordinate = 9 feet

Drag factor = 1.13

Skid speed = (70 x 70 x 1.13) / (2 x 9)

Skid speed = 1391 / 18

Skid speed ≈ 77.3 feet per second

To convert the speed to miles per hour, we can multiply it by 3600/5280

= 77.3 x (3600/5280)

≈ 52.9 miles per hour

Therefore, the minimum speed at which the car was traveling is 52.9 miles per hour.

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Answer:

Done

Step-by-step explanation:

To solve the problem, we can use a Venn diagram to visualize the information provided in the question.

Let's draw three overlapping circles to represent the three groups of people: those who eat fruits, those who eat vegetables, and those who eat cheese.

We know that:

- 37 people eat fruits, so we write "37" in the circle representing fruits.

- 33 people eat vegetables, so we write "33" in the circle representing vegetables.

- 9 people eat both cheese and fruits, so we write "9" in the overlap between the cheese and fruits circles.

- 12 people eat both cheese and vegetables, so we write "12" in the overlap between the cheese and vegetables circles.

- 10 people eat both fruits and vegetables, so we write "10" in the overlap between the fruits and vegetables circles.

- 12 people eat only cheese, so we write "12" in the part of the cheese circle that does not overlap with the other circles.

- 3 people eat all three offerings, so we write "3" in the intersection of all three circles.

Here's what the Venn diagram looks like:

```

Cheese

+----9----+----3----+----12---+

| | Cheese | | |

| Fruits +----6----+--4 | |

| | | | |

+----10---+----3----+----8---+---+

| Vegetables |

+------------+

9 24 15

```

To find out how many people eat cheese, we add up the numbers in the cheese circle: 9 + 3 + 12 = 24. So, 24 people eat cheese.

To find out how many people do not eat any of the offerings, we need to add up the numbers in the regions that are not part of any circle: the region outside of all circles, and the region in the center of the Venn diagram. These regions add up to 9 + 15 = 24. So, 100 - 24 = 76 people do eat at least one of the offerings.

The value of c for same sign of roots are,

⇒ c > 0

Given that;

Quadratic equation is,

⇒ x² - 2x + c = 0

Since, We know that;

For the roots of ax²+bx+c=0 to have same signs,

a(x2+b/ax+c/a), the last term, i.e. c/a>0, because if you factorize the quadratic, to arrive at positive constant you either have to have two negative numbers multiplied or two positive multiplied by each other.

Here,

Quadratic equation is,

⇒ x² - 2x + c = 0

Hence, The condition for same sign of roots are,

⇒ c > 0

Thus, The value of c for same sign of roots are,

⇒ c > 0

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Answer:

2. (i)  p = 4

   (ii)  p < -4

   (iii)  0 < c ≤ 1

Step-by-step explanation:

To find the value(s) of p for which x² - 2x - 3 = p will have the equal or real roots, we can use the discriminant formula.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Discriminant}\\\\$b^2-4ac$ \quad when $ax^2+bx+c=0$\\\\when $b^2-4ac > 0 \implies$ two real roots.\\when $b^2-4ac=0 \implies$ one real root (equal roots).\\when $b^2-4ac < 0 \implies$ no real roots (two complex roots).\\\end{minipage}}[/tex]

Rearrange the given equation so that it is in the form ax² + bx + c = 0:

[tex]x^2-2x-3-p=0[/tex]

Therefore, the values of a, b and c are:

To find the value of p for which the given quadratic has equal roots, substitute the values of a, b and c into the discriminant formula, set it equal to zero, and solve for p:

[tex]\begin{aligned}b^2-4ac&=0\\(-2)^2-4(1)(-3-p)&=0\\4-4(-3-p)&=0\\4+12+4p&=0\\16+4p&=0\\4p&=-16\\p&=-4\end{aligned}[/tex]

Therefore, the value of p for which the given quadratic will have equal roots is p = -4.

Rearrange the given equation so that it is in the form ax² + bx + c = 0:

[tex]x^2-2x-3-p=0[/tex]

Therefore, the values of a, b and c are:

To find the values of p for which the given quadratic has no real roots, substitute the values of a, b and c into the discriminant formula, set it to less than zero, and solve for p:

[tex]\begin{aligned}b^2-4ac& < 0\\(-2)^2-4(1)(-3-p)& < 0\\4-4(-3-p)& < 0\\4+12+4p& < 0\\16+4p& < 0\\4p& < - 16\\p& < -4\end{aligned}[/tex]

Therefore, the value of p for which the given quadratic will no real roots is p < -4.

For the roots of x² - 2x + c = 0 to have the same sign, both roots either have to be less that 0 or more than 0.

From part (i), we know that x² - 2x - 3 - p = 0 has equal roots when p = -4.

Substitute p = -4 into the equation:

[tex]\begin{aligned}x^2 - 2x - 3 - p &= 0\\x^2-2x-3-(-4)&=0\\x^2-2x-3+4&=0\\x^2-2x+1&=0\end{aligned}[/tex]

Comparing this to x² - 2x + c = 0, we can see that x² - 2x + c = 0 has equal roots when c = 1.

The leading coefficient of x² - 2x + c = 0 is positive, so the parabola opens upwards. We know it has equal roots when c = 1, so the vertex touches the x-axis when c = 1. Therefore, it has no real roots when c > 1 (since the vertex will be above the x-axis in this interval).

Therefore, we can determine that x² - 2x + c = 0 has two real roots when c ≤ 1.

When c = 0, the equation of the parabola is y = x² - 2x.

The roots are the points at which the curve crosses the x-axis (when y = 0). As y = 0 when x = 0, one of the roots is (0, 0) when c = 0.

Therefore, when c < 0, one root will be negative and the other will be positive.

Note there is no value of c where both roots are negative, as the x-value of the vertex is positive.

So the values of c for which the roots of x² - 2x + c = 0 will have the same sign (positive) are 0 < c ≤ 1.