Let c be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. find the exact length of c from the origin to the point 4, 8, 32 3 .

Answers

Answer 1

Let [tex]x=t[/tex], so that

[tex]x^2=2y\implies t^2=2y\implies y=\dfrac{t^2}2[/tex]

[tex]3z=xy\implies3z=\dfrac{t^3}2\impiles z=\dfrac{t^3}6[/tex]

Then the length of the path is

[tex]\displaystyle\int_{t=0}^{t=4}\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2+\left(\frac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^4\sqrt{1+t^2+\frac{t^4}4}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^4\sqrt{4+4t^2+t^4}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^4\sqrt{(t^2+2)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^4(t^2+2)\,\mathrm dt[/tex]
[tex]=\dfrac{44}3[/tex]

Answer 2

Final answer:

To find the exact length of the curve c from the origin to a given point, we can parameterize the curve and use the formula for arc length. In this case, we need to solve for x, y, and z in terms of a parameter t. Then, we can integrate the arc length formula to obtain the exact length.

Explanation:

To find the exact length of the curve c from the origin to the point (4, 8, 32/3), we need to first parameterize the curve. Given that x2 = 2y and 3z = xy, we can solve for x, y, and z in terms of a parameter t. Then, we can use the formula for arc length to calculate the length of the curve using integration.

Let's begin by solving for x, y, and z in terms of t:

From x2 = 2y, we have x = √(2t) and y = t2/2.Substituting these values into 3z = xy, we get 3z = √(2t) * (t2/2) or z = (t3√2)/6.

Now, we can calculate the arc length using the formula:

L = ∫ sqrt((dx/dt)2 + (dy/dt)2 + (dz/dt)2) dt

Plugging in the values of x, y, and z in terms of t, we can evaluate the integral to find the exact length of curve c.

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Related Questions

Madison deposited a total of $5,000 between two saving accounts bearing simple interest. One of the accounts has an interest rate of 4% while the other rate is 5%. If the total interest earned after one year is $220, find the amount deposited into each of the accounts.

Answers

0.04x+0.05 (5000-x)=220
Solve for x to find the amount invested at 4%
0.04x+250-0.05x=220
0.04x-0.05x=220-250
-0.01x=-30
X=30÷0.01
X=3,000 invested at 4%

5000-3000=2000 invested at 5%

Find the area of the shaded region. The units are in millimeters.
Use 3.14 for π. Round your answer to two decimal places.

Answers

Area of square=6*6=36mm^2
Area of circle=pi*r^2
The radius here is 3 because the diameter is 6.
3.14*3^2
3.14*9=28.26mm^2
The shaded region is 36-28.26=7.74
Final answer: 7.74 mm^2

Davis Construction is building a new housing development. They begin by mapping the development out on a coordinate grid. They place one of the two swimming pools at coordinate (-16, -25) and the other at (18, 23). If they want the club house to be exactly halfway between the two pools, find the coordinate point for the clubhouse

Answers

To solve this problem, we can use the following ratio formulas to find for the coordinates of the club house:

x = [m / (m + n)] (x2 – x1) + x1

y = [m / (m + n)] (y2 – y1) + y1

Where,

m and n are the ratios (0.5 and 0.5 each since the club house are to be located exactly halfway)

x1 and y1 are the x and y coordinates of the 1st pool (-16, -25)

x2 and y2 are the x and y coordinates of the 2nd pool (18, 23)

Substituting:

x = [0.5 / (0.5 + 0.5)] (18 - -16) + -16

x = 1

y = [0.5 / (0.5 + 0.5)] (23 - -25) + -25

y = -1

Therefore the club house should be located at coordinates (1, -1).

The area of a rectangular patio is 5 5/8 square yards, and it's length is 1 1/2 yards. What is the patio's width in yards?

Answers

Hello there! Thank you for asking your question here at Brainly. I will be assisting you today with answering this problem, and will be teaching you how to deal with it on your own in the future.

First, let's take a look at our question, and evaluate it.

"The area of a rectangular patio is 5 5/8 square yards, and it's length is 1 1/2 yards. What is the patio's width in yards?"

To clarify this problem, we are looking for the patio's width.

Let's first understand what the area of a rectangular shape is.
The formula for the area of a rectangle is "Length times Width", or "L • W".

So this is how the equation should look like:
A = L • W
We have our area, 5 5/8, and we have our length, 1 1/2.

To make things more simple, let's convert our fractions to decimals. Now, to convert our fractions to decimals, let's set our denominators (the numbers on the bottom of a fraction) equal to another fraction, with x as the numerator (the numbers on the top of a fraction) and 100 as the denominator.

So we have 1/2 and x/100. Divide 100 by 2 to find x (as 1/2 of anything is dividing by 2).
100 / 2 = 50, so 1 1/2 = 1.50 as a decimal.

Now, let's try 5/8.
1/8 = 0.125, so multiply 0.125 by 5.
0.125 • 5 = 0.625.
5 5/8 = 5.625 as a decimal.

So, now we have our equation:
A = L • W
Plug in our numbers.
5.625 = 1.50x

To isolate and solve for x, we need to divide both sides by 1.50, so let's do that.
5.625 / 1.50 = 3.75
1.50x / 1.50 = x

We are now left with:
x = 3.75

Your answer is:
The patio's width is 3.75 yards.

I hope this helps!

The manager at an ice cream shop keeps track of the number of deluxe cones and regular cones sold each day and the total money received. On Saturday, a total of 95 cones were sold, and the money collected was $628 . If deluxe cones are sold for $8 and regular cones are sold for $5 , how many deluxe cones and regular cones were sold?

Answers

d + r = 95.....d = 95 - r
8d + 5r = 628

8(95 - r) + 5r = 628
760 - 8r + 5r = 628
-8r + 5r = 628 - 760
-3r = -132
r = -132/-3
r = 44 <== regular cones

d + r = 95
d + 44 = 95
d = 95 - 44
d = 51 <== deluxe cones

Final answer:

By forming a system of equations and applying the elimination method, we conclude that the ice cream shop sold 51 deluxe cones and 44 regular cones on Saturday.

Explanation:

How to Solve for the Number of Deluxe and Regular Cones Sold

To solve for the number of deluxe cones and regular cones sold at the ice cream shop, we need to set up a system of equations. We are given that a total of 95 cones were sold and the total money collected was $628. Deluxe cones are sold for $8 and regular cones for $5. Let's define our variables:

x = number of deluxe conesy = number of regular cones

Now, we can establish our equations:

The total number of cones sold is 95: x + y = 95The total money collected is $628: 8x + 5y = 628

To solve the system, we can use the substitution or elimination method. Let's use elimination for this example.

Multiply the first equation by 5:

5x + 5y = 475

Now subtract this from the second equation:

(8x + 5y) - (5x + 5y) = 628 - 4753x = 153x = 51

With 51 deluxe cones sold, we can substitute that into the first equation to find y:

51 + y = 95y = 95 - 51y = 44

So, 51 deluxe cones and 44 regular cones were sold on Saturday.

Two garden plots are to have the same area. One is square and one is rectangular. The rectangular plot is 2 meters wide and 18 meters long. How long is one side of the of the square garden plot in meters?

Answers

area of rectangle = l x w

area of rectangle = 2*18 = 36 square meters

area of square = S^2, or s = square root(area)

sqrt(36) = 6

length of side of square = 6 meters

The side length of the square garden plot with the same area as a rectangular plot of dimensions 2 meters by 18 meters is 6 meters.

The student is asking about finding the length of a side of a square garden plot that has the same area as a rectangular garden plot measuring 2 meters wide by 18 meters long. To find the area of the rectangular plot, we multiply the length by the width: 2 meters imes 18 meters = 36 square meters.

Since the area of the square plot must equal the area of the rectangular plot, we can use the formula for the area of a square, A = s2, where s is the side length of the square. Therefore, to find the side length of the square plot, we take the square root of the area of the rectangular plot:
√(36 square meters) = 6 meters

So, the side length of the square garden plot is 6 meters.

2(8+h) simplify using distrubtive property

Answers

Hello there!

2(8 + h)
The Distributive Property allows us to multiply any number/variable outside of the parenthesis by all numbers/variables inside the parenthesis.

For example;
2(1 + 2)
2(1) + 2(2)
2 + 4
6.

Let's try this with our current problem.
2(8 + h)
2(8) + 2(h)
16 + 2h (This is your simplified expression)

I hope this helps!

4 students equally share 3/4 of a pizza. How much of the pizza does each student get?

Answers

16/3 of the pizza

3/4 divided by four means you have to take the reciprocal of 3/4 to give you 4/3 and then multiply that by 4 to get 16/3.

Final answer:

To find out how much pizza each of 4 students gets when they equally share 3/4 of a pizza, we divide the amount of pizza (3/4) by the number of students (4). The result is 3/16, which means each student gets 3/16 of the pizza.

Explanation:

The question asks how 4 students can equally share 3/4 of a pizza. In mathematics, to divide a quantity equally amongst a group, we perform division. So, in this scenario, we need to divide 3/4 (the pizza) by 4 (the students).

Step 1: Represent the given scenario as a division problem: 3/4 ÷ 4. Step 2: Convert the division into multiplication by the reciprocal. This changes our problem into a multiplication problem: 3/4 * 1/4. Step 3: Multiply the numerators together to get the numerator of the answer: 3 * 1 = 3. Step 4: Multiply the denominators together to get the denominator of the answer: 4 * 4 = 16.

So, the answer of 3/4 ÷ 4 is equal to 3/16. Therefore, each student gets 3/16 of the pizza.

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someone please help me to find the answer to this geography problem please explain

Answers

First, we must know what the two variables in the equation y=mx+b really means.

The M represents slope while the B represents the Y-intercept.

With that out of the way, take a look at your top equation, y = 3/2x + ?. If you noticed the question mark is in the spot of the B (Y-intercept), we can simply find this by finding where on the graph does the line intercept the Y-axis.

Triangle B shows the Y-intercept to be -7, this means that the hypotenuse intercepts the Y-axis at (0 , -7).

If we were to find the (what I like to call) the interception point of triangle A's hypotenuse and the Y-axis, we find our missing variable.

Just looking at the graph, we can count and visually determine that the Y-intercept of the hypotenuse is (0 , 1).

Therefore, we can say: The equation of A's hypotenuse is:
y = 3/2x + 1

Hope I could help you out!
If my answer is incorrect, or I provided an answer you were not looking for, please let me know. However, if my answer is explained well and correct, please consider marking my answer as Brainliest! :)
Have a good one.God bless!

A set of data with a mean of 45 and a standard deviation of 8.3 is normally distributed. Find the value that is +1 standard deviation away from the mean.

Answers

Answer:

53.3

Step-by-step explanation:

Add one standard deviation to the mean:

45 +1(8.3) = 53.3

Answer:

53.3

Step-by-step explanation:

+1 standard deviation away from the mean is 45+1(8.3)

which algebraic expression represents the phase

Answers

The keyword here is product.

A product is the result when multiplying.

Therefore, the answer here is B.

Use appropriate algebra and theorem 7.2.1 to find the given inverse laplace transform. (write your answer as a function of t.) −1 s (s − 5)(s − 6)(s − 30)

Answers

When it comes to laplace equations, there are transformation equations to follow. Generally, when you want to transform a laplace equation, you change the equation from f(t) to F(s). If you do the reverse, it is called the reverse laplace equation.

Based on the given, the useful transformation equation is shown in the attached picture. The given equation is

[tex] \frac{-1}{s(s-5)(s-6)(s-30)} [/tex]

Breaking it into partial fractions,

[tex] \frac{-1}{s} * \frac{1}{s-5} * \frac{1}{s-6} * \frac{1}{s-30} [/tex]

From the given transformation equation:

-1/s ⇔ -e^0t ⇔ 1
1/(s-5) ⇔ e^-(-5t) ⇔ e^5t
1/(s-6) ⇔ e^-(-6t) ⇔ e^6t
1/(s-30) ⇔ e^-(-30t) ⇔ e^30t

Therefore, the laplace equation is equal to 1*e^5t*e^6t*e^30t which is simplified into e^41t,

If all of the diagonals are drawn from a vertex of a pentagon, how many triangles are formed?

Answers

A picture helps. There are 2 non-adjacent vertices, so two diagonals can be drawn, resulting in 3 triangles.

Drawing all diagonals from one vertex of a pentagon to the non-adjacent vertices creates three triangles within the pentagon.

When all the diagonals are drawn from one vertex of a pentagon, it divides the pentagon into several triangles.

A pentagon has five sides, and when selecting a single vertex, we can draw diagonals to the non-adjacent vertices. In this case, there are three non-adjacent vertices, which means we can draw three diagonals, thus creating three triangles inside the pentagon.

This is because each diagonal, combined with two sides of the pentagon, forms a triangle.

Calculus AB
lim x-->4 sqrt(x+5)-3 / 4-x
The answer is apparently -1/6
How do you solve this

Answers

assuming you mean
[tex] \lim_{x \to 4} \frac{\sqrt{x+5}-3}{4-x}[/tex]

that means as x approaches 4

if we sub 0 for x we get
0/0
and intermitent form
use l'hopital's rule

so
take the derivitive of the top and bottom seperatly
l'hopitals rule is something like
if [tex] \lim_{x \to n} \frac{f(x)}{g(x)}[/tex] results in 0/0 or -∞/∞ or∞/∞ then keep doing it until f(n)/g(n) gives a form that isn't intermitent

so

take derivitive of top and bottom
[tex]\dfrac{\frac{1}{2\sqrt{x+5}}}{-1}[/tex]
now, if we subsitute 4 for x we get
[tex]\dfrac{\frac{1}{2\sqrt{4+5}}}{-1}[/tex]=

[tex]\dfrac{\frac{1}{2\sqrt{9}}}{-1}[/tex]=

[tex]\dfrac{\frac{1}{2(3)}}{-1}[/tex]=

[tex]\dfrac{\frac{1}{6}}{-1}[/tex]=

[tex]\dfrac{1}{-6}=\dfrac{-1}{6}[/tex]

John predicts the basketball team will score 68 points. If the basketball team actually scores 49 points, what is John's percent error? Round your answer to the nearest tenth of a percent.

Answers

%error=100(estimated-actual)/actual

%error=100(68-49)/49

%error≈38.8%

Since this is positive, John's error overestimated the teams score by about 38.8%

The pressure,p, of a gas varies inversely with its volume,v. Pressure is measured in units of pa. Suppose that a particular amount of a gas is initially at a pressure of 104 pa at a volume of 108 L. If the volume is expanded to 432 L, what will the new pressure be

Answers

The pressure of a gas varies inversely with its volume.

In this case, the gas have 108L volume in 104pa pressure and you asked how much the pressure if the gas made into 432L volume. Since the volume is bigger it should be clear that the pressure would be lower.
The equation of the new pressure would be : 104pa x (108L / 432L) = 104pa x 0.25= 26pa

Answer: 26pa

Suppose that a particular artillery piece has a range r = 8320 yards . find its range in miles. use the facts that 1mile=5280ft and 3ft=1yard.

Answers

The range in miles is 4.72 mile

What do you mean by 1 mile?

mile, any of various units of distance, such as the statute mile of 5,280 feet (1.609 km). It originated from the Roman mille passus, or “thousand paces,” which measured 5,000 Roman feet.

Given : range= 8320 yards

1 mile= 5280 feet

3 feet= 1 yard

Now,

8320yards X 3ft = 24,960

Now, divide that by 5,280ft in a mile

= 24960/5280

= 4.72 miles

Hence, range is 4.72 miles.

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Final answer:

To convert the artillery piece's range of 8320 yards to miles, you multiply by 3 to change yards to feet and then divide by 5280 to convert feet to miles, resulting in a range of approximately 4.727 miles.

Explanation:

To find the artillery piece's range in miles, starting with its range in yards, we need to use unit conversion factors. Knowing that 1 mile equals 5280 feet and 3 feet equals 1 yard, we can set up a conversion to calculate the distance in miles.

Steps to Convert Yards to Miles:

Convert the given distance from yards to feet by multiplying the number of yards by 3 (since there are 3 feet in a yard).Next, convert the distance from feet to miles by dividing the number of feet by 5280 (since there are 5280 feet in a mile).

For an artillery range of 8320 yards:

8320 yards is equivalent to 8320 yards * 3 feet/yard = 24960 feetThen 24960 feet / 5280 feet/mile gives us the range in miles.

When you divide 24960 by 5280, you get 4.727 miles (rounded to three decimal places).

Pick the correct description of the line 5x + 3y=15

Answers

5x + 3y = 15
3y = -5x + 15
y = -5/3x + 5

slope is -5/3
y int (where ur line crosses the y axis) is (0,5)
x int (where ur line crosses the x axis) is (3,0)

ur line will be slanting downward because of ur negative slope

Answer:

The correct description is [tex]y=5-\frac{5x}{3}[/tex]

Step-by-step explanation:

In this case, firts we have to organize the equation in the normal way, where y is alone in one side, this is:

[tex]5x+3y=15\\3y=15-5x\\y=\frac{15}{3}-\frac{5x}{3} \\y=5-\frac{5x}{3}[/tex]

The correct description is [tex]y=5-\frac{5x}{3}[/tex]

Now we can see that the slope of the equation is [tex]-\frac{5}{3}[/tex]

0.025 in scientific notation (step by step)

Answers

First, find out how many place values you must move the decimal in order to have it right after the first non-zero digit.
In this case, you must move the decimal to the right by 2 place values.
Then, write the number again, this time with the decimal placed after the first nonzero digit. In this case, write 2.5
In scientific notation, you have to multiply that number by 10^x
The x value is the number of place values you moved. In this case it is 2.
If you moved to the right, add a negative sign before it. If you moved the decimal to the left, keep it as a positive value.
Final answer: 2.5 x 10^-2

0.025 in scientific notation is 2.5 × 10⁻².

To express 0.025 in scientific notation, we need to convert the decimal form to a number multiplied by a power of 10.

Here's how you can do it step by step:

Start with the given decimal number: 0.025.

Identify the non-zero digit. In this case, it is 2.

Move the decimal point to the right until there is only one non-zero digit to the left of the decimal point. Count the number of places you moved the decimal point. In this case, we moved the decimal point two places to the right.

0.025 -> 2.5

Write the non-zero digit (2.5) as a decimal number.

Finally, express the power of 10 that represents the number of places you moved the decimal point. Since we moved the decimal point two places to the right, the power of 10 is -2.

Putting it all together, we have:

0.025 = 2.5 × 10⁻²

Therefore, 0.025 in scientific notation is 2.5 × 10⁻².

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What is the difference of the polynomials?

(8r6s3 – 9r5s4 + 3r4s5) – (2r4s5 – 5r3s6 – 4r5s4)

Answers

8r^6s^3 - 9r^5s^4 + 3r^4s^5 - (2r^4s^5 - 5r^3s^6 - 4r^5s^4) =
8r^6s^3 - 9r^5s^4 + 3r^4s^5 - 2r^4s^5 + 5r^3s^6 + 4r^5s^4 =
8r^6s^3 -5r^5s^4 + r^4s^5 + 5r^3s^6 <==

Answer: The required difference is [tex]8r^6s^3-5r^5s^4+r^4s^5+5r^3s^6.[/tex]

Step-by-step explanation: We are given to find the difference of the polynomials as follows :

[tex]DP=(8r^6s^3-9r^5s^4+3r^4s^5)-(2r^4s^5-5r^3s^6-4r^5s^4)~~~~~~~~~~~~~~~~~~~(i)[/tex]

To find the given difference, we need to choose the terms with same poers of the unknown variables r and s and then subtract their coefficients.

The difference (i) is as follows :

[tex]DP\\\\=(8r^6s^3-9r^5s^4+3r^4s^5)-(2r^4s^5-5r^3s^6-4r^5s^4)\\\\=8r^6s^3-9r^5s^4+3r^4s^5-2r^4s^5+5r^3s^6+4r^5s^4\\\\=8r^6s^3+(4-9)r^5s^4+(3-2)r^4s^5+5r^3s^6\\\\=8r^6s^3-5r^5s^4+r^4s^5+5r^3s^6.[/tex]

Thus, the required difference is [tex]8r^6s^3-5r^5s^4+r^4s^5+5r^3s^6.[/tex]

Find the area of a square with radius 17√2 in. Round to he nearest whole number. Please help me!!

Answers

The whole diagonal is [tex]34 \sqrt{2} [/tex] which makes us an equation : [tex]x \sqrt{2} = 34 \sqrt{2} [/tex] so each side is 34! Then we can get the area by doing [tex]34^{2} = 1156[/tex]! The answer is A!

The area of the square with radius 17√2 is 1156 square inches.

What is area?

The area is the amount of space within the perimeter of a 2D shape

What is the formula for the area of square?

The formula for the area of square is

[tex]A = \frac{1}{2} d^{2}[/tex]

Where, d is the length of the diagonal of the square.

According to the given question.

The radius of the square is [tex]17\sqrt{2}[/tex] in.

⇒ Diameter or the length of the square = 2 × 17 √2 = 34√2

Therefore,

The area of the square

= [tex]\frac{1}{2}( 34(\sqrt{2)} )^{2}[/tex]

= [tex]\frac{2312}{2}[/tex]

=[tex]1156\ in^{2}[/tex]

Hence, the area of the square with radius 17√2 is 1156 square inches.

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A rectangle is 6 centimeters longer than is wide and has an area of 280 square centimeters if l represent the length,in centimeters of the rectangle solving which of the following formulas will give the correct value for l?

Answers

A = L * W
W = L - 6

280 = L(L - 6) or 280 = L^2 - 6L

l = w+6
a = 280 = w*(w+6)
280 = w^2+ 6w.
0 = w^2 + 6w - 280
w^2 + 20w - 14w - 280
(w^2 + 20w) - (14w-280)
w(w + 20) - 14(w + 20)
0 = w - 14 OR 0 = w + 20
-20=w OR 14=w
14 must be the answer, you cannot have negative width. Ergo, length = 20.
Width + 6, 14 + 6, 20 = length

which property of congruence would prove that BD ≈ BD?

the reflexive property of congruence
the symmetric property of congruence
the transitive property of congruence
None of the choices are correct.

Answers

It appears to be reflexive, since it's an angle congruent to itself.

The function g(x) = 3(x - 3)2 + 2 will translate 3 units to the right. True or False?

Answers

The answer is true, since horizontal shifts are shown in a function by either adding or subtracting a value from the variable. The horizontal shift is (x-3), not (x+3) since in order for (x+3) = 0, x has to equal -3. The x-intercept, if not for the vertical shift of 2, would be 3, not -3, meaning (x-3) is the correct way to represent translating 3 units to the right.

Which figure has the correct lines of symmetry drawn in?

Answers

The answer is the fourth image.

If you were to fold the hexagon in half, it would be symmetrical when folded from vertex-to-vertex and when folded from the midpoint of a side to the opposite side.

This means that while the first and third images are valid lines of symmetry, they are incomplete because they're missing the second set of lines from side-to-side.
The second image also has valid lines of symmetry, but it doesn't have all of them.

Thus, the answer is the fourth image.

Find the work done by a force fâ =1.8i^+2.2j^n as it acts on an object moving from the origin to the point râ =56i^+31j^m.

Answers

Take the dot product:

[tex]W=\mathbf F\cdot\mathbf r=(1.8\,\mathbf i+2.2\,\mathbf j)\cdot(56\,\mathbf i+31\,\mathbf j)[/tex]
[tex]W=169\,\text{Nm}[/tex]

Final answer:

The work done by the force is found by taking the dot product of the force vector and the position vector. This is achieved by multiplying corresponding components of the vectors and then adding the results together.

Explanation:

To find the work done by the force, we use the formula Work = Force . Displacement, where the '.' signifies a dot product. In this case, the force vector is fâ = 1.8i^ + 2.2j^ and the position vector, râ = 56i^ +31j^. The dot product between force fâ and the displacement (position vector) râ is calculated by multiplying corresponding i and j components and then adding them together. Therefore, Work = (1.8i^.56i^) + (2.2j^.31j^). Perform the multiplication for both components and then add the results to get the total work done by the force.

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What is the value of of angle BAD:

30◦
5◦
15◦
20◦

Answers

From 2x+5=3x, then x=5.

Helpp! -- A piece of wire 5 inches long is to be cut in two pieces. One piece is x inches long and is to be bent into the shape of a square. The other piece is to be bent in the shape of a circle. Find an expression for the total area made up by the square and the circle as a function of x.

Answers

check the picture below.

since, we know that for the circle, the perimeter is s+s+s+s or 4s, then 4s = x and so on.

now, for the circle, the perimeter is just the circumference, circumference of a circle is 2πr, thus 2πr = 5-x and so on.

anyway... so... the equation of the area made by those two in x-terms, is just their sum... so let's do that then

[tex]\bf \pi \left( \cfrac{5-x}{2\pi } \right)^2\quad +\quad \cfrac{x^2}{4^2}\\\\ -------------------------------\\\\ \pi\cdot \cfrac{(5-x)^2}{(2\pi )^2}+\cfrac{x^2}{4^2}\implies \cfrac{\pi (5-x)^2}{2^2\pi^2}+\cfrac{x^2}{4^2}\implies \cfrac{(5-x)^2}{4\pi}+\cfrac{x^2}{4^2} \\\\\\ \textit{let's use the LCD of }4^2\pi \textit{ to add them up} \\\\\\ \cfrac{4(5-x)^2+\pi x^2}{4^2\pi }\implies \cfrac{4(5-x)^2+\pi x^2}{16\pi }[/tex]

now, you can expand the squared binomial on the numerator, but there will not be any like-terms to simplify, so, it won't make much difference, so.... you can expand it or not.

Final answer:

The expression for the total area made up by the square and the circle as a function of x is f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) square inches.

Explanation:

To solve this problem, you need to know the formulas for the areas of a square (Area = side^2) and a circle (Area = pi * radius^2). If a piece of the wire with length x inches is to be bent into the shape of a square, then each side of the square will be x/4 inches long (since a square has 4 equal sides). Therefore, the area of the square = (x/4)^2 = x^2/16 sq.inches.

The remaining piece of the wire is 5-x inches long and is to be bent into the shape of a circle. The circumference of the circle is 5-x inches (since it uses the remaining wire), therefore the radius of the circle is (5-x)/(2*Pi) inches. Therefore, the area of the circle = Pi*((5-x)/(2*Pi))^2 = (25 - 10x + x^2)/(4*Pi) sq.inches.

The total area made up by the square and the circle is the sum of their areas, therefore the function representing the total area f(x) = x^2/16 + (25 - 10x + x^2)/(4*Pi) sq.inches.

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The formula can be used to find the velocity v in feet per second of an object that has fallen h feet. Find the velocity of an object that has fallen 23 feet. Round your answer to the nearest hundredth.

Answers

The formula for a falling object is
v = u +gt
where
u = initial velocity (=0 for a falling object)
g = acceleration due to gravity, 32.2 ft/s²
t = time, s

In terms of distance fallen,
v² = u² + 2gh
where h = distance fallen, ft.

Because h = 23 ft and u = 0, therefore
v² = 2*32.2*23 = 1481.2
v = 38.486 ft/s

Answer: 28.49 ft/s (nearest hundredth)

James wants to tile his floor using tiles in the shape of a trapezoid. To make the pattern a little more interesting he has decided to cut the tiles in half along the median. The top base of each tile is 15 inches in length and the bottom base is 21 inches. How long of a cut will John need to make so that he cuts the tiles along the median?

Answers

Answer:18 inches

Step-by-step explanation:

The answer would be 18 inches.

What is the median of a trapezoid?

The median of a trapezoid is the segment that connects the midpoints of the non-parallel sides.

The length of the median is the average of the length of the bases.

The top base of each tile is 15 inches in length and the bottom base is 21 inches,

Add the top base and bottom base,

So, 15+21=36

Now, divide that by 2

⇒ 18

Hence, the answer would be 18 inches.

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