A square with an area of 169 cm^2 is rotated to form a cylinder. What is the height of the cylinder?

Let p(x) be a polynomial, and suppose that a is any real number. Prove that

lim x→a p(x) = p(a) .

Solution. Notice that

2(−1)4 − 3(−1)3 − 4(−1)2 − (−1) − 1 = 1 .

So x − (−1) must divide 2x^4 − 3x^3 − 4x^2 − x − 2. Do polynomial long division to get 2x^4 − 3x^3 − 4x^2 – x – 2 / (x − (−1)) = 2x^3 − 5x^2 + x – 2.

Let ε > 0. Set δ = min{ ε/40 , 1}. Let x be a real number such that 0 < |x−(−1)| < δ. Then |x + 1| < ε/40 . Also, |x + 1| < 1, so −2 < x < 0. In particular |x| < 2. So

|2x^3 − 5x^2 + x − 2| ≤ |2x^3 | + | − 5x^2 | + |x| + | − 2|

= 2|x|^3 + 5|x|^2 + |x| + 2

< 2(2)^3 + 5(2)^2 + (2) + 2

= 40

Thus, |2x^4 − 3x^3 − 4x^2 − x − 2| = |x + 1| · |2x^3 − 5x^2 + x − 2| < ε/40 · 40 = ε.

Final answer:

The tree in Carlos' backyard is 5 meters high, which is equivalent to 500 centimeters because there are 100 centimeters in a meter. Multiplying 5 meters by 100 gives us the height in centimeters.

Explanation:

To convert the height of the tree in Carlos' backyard from meters to centimeters, we need to know the conversion factor between these two units of measurement. There are 100 centimeters in a meter. Therefore, if the tree is 5 meters high, to find its height in centimeters, we multiply 5 meters by 100.

5 meters × 100 centimeters/meter = 500 centimeters

So, the tree is 500 centimeters tall when converted from meters. This is similar to the example given where Corey measures a distance of 8 meters between two trees and wants to convert that measurement to centimeters. In this concept, we are provided insight into converting with metric units of measurements which is crucial for accurately understanding dimensions in different units.

Cory's percent error for his trip time prediction was 13.5% after rounding to the nearest tenth of a percent.

To calculate Cory's percent error for predicting his travel time to Baltimore from Washington D.C., we can use the formula for percent error:

Percent Error =[tex]|(Actual Value - Predicted Value) / Actual Value| \times 100%[/tex]

In this case, Cory's predicted time (Predicted Value) is 64 minutes, and the actual time taken (Actual Value) is 74 minutes. We can plug these values into the formula:

Percent Error = [tex]|(74 - 64) / 74| \times 100%[/tex]
Percent Error = [tex]|10 / 74| \times 100%[/tex]
Percent Error =[tex]0.1351 \times 100%[/tex]

Calculating further:

Percent Error = 13.51%

When rounded to the nearest tenth, Cory's percent error is 13.5%.

The polynomial in standard form from the given options is [tex]x^4y^2 + 4x^3y + 10x[/tex], as it orders the terms by degree in descending order, first by the degree of x and then y.

The term standard form in mathematics, especially in relation to polynomials, refers to a way of writing the polynomial so that the terms are ordered by their degree in descending order. More specifically, for a polynomial in two variables, x and y, the standard form would have the terms arranged first by the degree of x, then y, from highest to lowest.

Looking at the options provided in the question, the polynomial that is written in standard form would have the highest degree term first, and so on. The polynomial [tex]x^4y^2 + 4x^3y + 10x[/tex] follows this convention, with the terms ordered by decreasing powers of x first and y second. Therefore, this is the polynomial written in standard form.

Note that while the other options are all polynomials, they do not follow the standard form convention as closely as the correct option provided.

Answer:  Equation relating C to S   : [tex]C=6500+250S[/tex]

The total cost to transport 19 tons of sugar is $11, 250.

Step-by-step explanation:

Given : It will cost $6500 to rent trucks, and it will cost an additional $250 for each ton of sugar transported.

Total cost =  $6500 + $250 x (amount of sugar transported ( in tons))

Let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported.

Then, the equation relating C to S  would be : [tex]C=6500+250S[/tex]

When S= 19 , we get

[tex]C=6500+250(19)=6500+4750= 11250[/tex]

Hence, the total cost to transport 19 tons of sugar would be $11, 250.

An equation that relates Cost to the amount of sugar S

C = 6500 + 250S

The total cost is $1120

C = total cost

C = 6500 + 250(19)

Total cost = 6500 + 4750

= 11250

Therefore the total cost of transporting the 19 tons of sugar is $11250

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The value of the solution of expression is, x = 6

We have to give that,

An expression to simplify,

5x - 7 = 3x + 5

Now, Simplify the expression by combining like terms as,

5x - 7 = 3x + 5

Subtract 3x on both sides,

5x - 3x - 7 = 3x + 5 - 3x

2x - 7 = 5

Add 7 on both sides,

2x - 7 + 7 = 5 + 7

2x = 12

Divide 2 into both sides,

x = 12/2

x = 6

Therefore, the solution is, x = 6

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total length of the line is:

 J = -10

K = 8

for a total of 18 units long

 midpoint would be 18/2 =9

-10+9 = -1

 so need the point that is located at negative 1, which is point N

 so N is the midpoint

The percentage increase of the notebook is 14.3%.

The percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.

Given that the price of a notebook was $3.50 yesterday. Today, the price rose to $4.00

The percent increase will be calculated as,

P = [ (4.00 - 3.50) / 3.50 ] x 100

P = [ 0.50 / 3.50 ] x 100

P = 0.1428 x  100

P = 14.28% or 14.3%

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Final answer:

After calculations, it is determined that Manuel drove approximately 269 miles.

Explanation:

Manuel rented a truck which had a base fee of $14.95 and an additional charge of 87 cents per mile. To find the number of miles driven, we need to subtract the base fee from the total amount paid and then divide by the cost per mile.

First, subtract the base fee from the total cost:

Total amount paid = $248.98

Base fee = $14.95

Amount paid for miles = $248.98 - $14.95 = $234.03

Next, divide the amount paid for miles by the cost per mile:

Cost per mile = 87 cents = $0.87

Number of miles driven = $234.03 / $0.87 = 269 miles (approximately)

Therefore, Manuel drove approximately 269 miles with the rented truck.

Water weighing 8.34 lb per gallon is equivalent to about 133.44 ounces per gallon.

The question at hand is how to convert the weight of water from pounds per gallon to ounces per gallon. Given that we know water weighs about 8.34 lb per gallon, we can use the conversion factor of 16 ounces in a pound to perform this calculation. Here's how you can do it:

Therefore, water weighs approximately 133.44 ounces per gallon.

To calculate when the height of the projectile will be 1538 feet, use the kinematic equation and solve the quadratic equation for time.

To calculate when the height of the projectile will be 1538 feet, we can use the kinematic equation for free-falling objects. The equation is: h = [tex]h0 + v0*t - 16*t^2,[/tex] where h is the height above ground, h0 is the initial height (0 in this case), v0 is the initial velocity (320 ft/sec in this case), and t is the time.

Substituting the given values into the equation, we have: 1538 = 0 + 320*t - 16*t^2. Rearranging this equation, we get: [tex]16*t^2[/tex]- 320*t + 1538 = 0.

Now we can solve this quadratic equation for t by using the quadratic formula: t = (-b ± sqrt([tex]b^2[/tex] - 4ac)) / (2a), where a = 16, b = -320, and c = 1538. Plugging in these values, we can calculate the values of t.

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The parametric vector equations for the parabola y=x² from the origin to the point (3,9) is x = t and y = t² for the parameter range 0 <= t <= 3.

The required parametric vector equation for the parabola y=x² from the origin to the point (3,9) is given by the equations x = t and y = t². Here, t is the parameter. At t=3, these equations give us the coordinates x=3 and y=9, which corresponds to the point (3,9). Thus, these equations accurately represent the parabola y=x² from the origin to the point (3,9) for the parameter range 0 <= t <= 3.

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In the parametric form, the path of a point on the parabola y=x² from the origin to point (3,9) is represented by a vector (t, t²) with t ranging from 0 to 3.

To find a vector parametric equation for the parabola y=x² from the origin to the point (3,9), we first need to understand that the parabola y = x² is not a vector in itself.

But, we can describe its trajectory through a parametric form using t as a parameter. For a given t, the parabola is represented by a vector (x(t), y(t)), where x(t) and y(t) are functions of t. For the specific parabola y = x², we can define the functions as x(t) = t and y(t) = t².

This means the vector at time t is given by (t, t²). From the origin (0,0) to the point (3,9), we now have a parameter t ranging from 0 to 3.

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In this case, the Poisson distribution is the best one to use. The formula for Poisson distribution is given as:

P[x] = e^-m * m^x / x! 

Where,

m = mean number of typographical errors = 6

x = sample value

A. The probability of exactly 4 errors are found on a page is:

P[4] = e^(-6) * 6^4/4!

P[4] = 0.1339

B. The probability that at most 4 errors will be the summation of x = 0 to 4:

P[0] = e^(-6) * 6^0/0! = 2.479 E -3

P[1] = e^(-6) * 6^1/1! = 0.01487

P[2] = e^(-6) * 6^2/2! = 0.04462

P[3] = e^(-6) * 6^3/3! = 0.08924

Therefore summing up all including the P[4] in A gives:

P[at most 4] = 0.2851

C. The probability that more than 4 would be the complement of answer in B.

P[more than 4] = 1 - P[at most 4]

P[more than 4] = 1 - 0.2851

P[more than 4] = 0.7149

Point Z's polar coordinates can be found by using the formulas x = r*cos(θ) and y = r*sin(θ), converting the angle from degrees to radians first. Use r = 11 and θ = -80 degrees.

This question deals with the concept of polar coordinates, coordinates given by a distance from the origin (radius) and an angle from the positive x-axis (-80 degrees in this case). Point Z's coordinates can be found using the formulas: x = r*cos(θ) and y = r*sin(θ). Here r (the radius) is 11  and θ is -80 degrees, but remember we need to convert this angle to radians because the trigonometric functions in most calculators use radians. That can be done using the formula: Radians = Degrees * (π / 180). Hence calculate x and y to find Point Z.

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To find the equivalent expression, multiply the first fraction by y and the second fraction by x. Simplify the expression by combining like terms.

To find the expression equivalent to the given complex fraction, we can simplify it step by step. First, multiply the numerator and denominator of the first fraction (2/x) by y, and multiply the numerator and denominator of the second fraction (-5/y) by x. This gives us (2y)/(xy) - (4x)/(-5x). Next, simplify the expression by combining like terms. The final equivalent expression is (2y - 4x)/(xy + 5x).

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After solving the equations, it is found that there are 32 pennies in the jar.

To solve the problem of determining the number of pennies in the jar, first we establish two variables: let's denote P for the number of pennies and N for the number of nickels.

According to the problem, there are 56 coins in total, which gives us the equation P + N = 56. Also, we know that the total value of the coins is $1.52, and because each penny is worth 1 cent and each nickel is worth 5 cents, we have another equation, which is P + 5N = 152 (since there are 100 cents in a dollar).

Now we have a system of two equations to work with:

P + N = 56

P + 5N = 152

To find P, we can subtract the first equation from the second equation to eliminate P and solve for N:
0P + 4N = 96

N = 96 / 4

N = 24

Now that we know there are 24 nickels, we can find the number of pennies by substituting N back into the first equation:

P + 24 = 56

P = 56 - 24

P = 32

Therefore, there are 32 pennies in the jar.