Find the curvature of r(t) = (3t, t^2, t^3) at the point (3, 1, 1).

Answers

Answer 1

The curvature of r(t) at the point (3,1,1) is 2/3

The curvature of a space curve is given by the formula:

k(t) = |r'(t) x r''(t)| / |r'(t)|^3 where,

r(t) is the position vector of the curver'(t) is the first derivative of r(t) r''(t) is the second derivative of r(t).

r(t) = (3t, t^2, t^3)

r'(t) = (3, 2t, 3t^2)

r''(t) = (0, 2, 6t)

So we can substitute these in the above formula:

k(t) = |(3, 2t, 3t^2) x (0, 2, 6t)| / |(3, 2t, 3t^2)|^3

By cross-product, we get

k(t) = |(6t, -6, -6t^2)| / |(3, 2t, 3t^2)|^3

Now substituting the point (3,1,1) we get

k(t) = |(6(1), -6, -6(1)^2)| / |(3, 2(1), 3(1)^2)|^3 = 6/9 = 2/3

So the curvature of r(t) at the point (3,1,1) is 2/3

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Related Questions

Find the equation of the line that passes through the point (6,-2) and is parallel to the line 5x+3y

Answers

Answer: y=(-5/3)x+8

Step-by-step explanation: earrange 5x+3y=6 in slope intercept form

y=mx=b

5x+3y=6

3y=6-5x

y=2-(5x)/3

then slope for a line parallel is -5/3

Use point slope form y-y1=m(x-x1)

plug un values of (x,y) = (6,-2)

and m=-5/3

y-y1=m(x-x1)

y-(-2)=(-5/3)(x-6)

y+2=(-5/3)x+10

y=(-5/3)x+10-2

y=(-5/3)x+8 <-- solution

Write an equation of the line that passes through each pair of points (9,2) (-2,6)

Answers

y = (-4/11)x + 5.27 is the equation that passes through each pair of points (9,2) (-2,6).

What is equation?

A group of numbers and variables without an equal sign are known as a numerical expression. A set of numbers and variables with an equal sign is referred to as an equation.

Once you are familiar with these terms, you will realise that an equation is just two expressions joined by an equal sign. You can also write expressions, equations, and simplify equations.

To find an equation of the line that passes through each pair of points (9,2) (-2,6) we need to find slope and convert it to y-intercept form

Slope formula = y - y₁ = m(x - x₁)

Putting the values in the formula we get

2 - 6 = m(9 - (-2))

-4 = m(11)

m = -4/11

As we have found slope, we can find the slope intercept form i.e.

y - 2 = -4/11(x -9)

y = (-4/11)x + 3.27 + 2

y = (-4/11)x + 5.27

Thus, y = (-4/11)x + 5.27 is the equation that passes through each pair of points (9,2) (-2,6)

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Which of the following is the product of the rational expressions shown
below?
2/x+1 • 5/3x

A. 5(x+1)/6x
B. 5/x^2+3
C. 10/3x+3
D. 10/3x^2+3x

Answers

Answer:

D. 10/3x^2+3x

Step-by-step explanation:

The product of the two rational expressions is:

(2/x+1) * (5/3x)

Multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second fraction:

(2*5) / (x+1) * (3x)

Which simplifies to:

10 / (x^2 + x)

So the correct answer is:

D. 10/3x^2+3x

Identify the value of x that makes each pair of ratios equal
5:15 and x:3

Answers

The ratio for 5 / 15 based on the information is 1:3. The value of x is 1.

How to illustrate the ratio?

Ratio in Mathematics simply demonstrates how many times one number can be able to fit into another number. It should be noted that ratios contrast two numbers by dividing them. In this case, A/B will be the formula if one is comparing one variable (A) to another variable (B).

This simply indicates that you're dividing the variable A by B. Let us say for instance, the ratio will be 3/10 if A is 3 and B is 10.

The ratio for 5 / 15 will be:

= 5 / 15

= 1 / 3

= 1:3

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For which addition equation can you make a 10 to add?
24+14=
7+25=
76+3=
26+14=

Answers

Nothing is 10 for this equation

Solve the following problem using the substitution method. Please make sure to show all of your work.

6x - 3y = 1
y = -2x - 5

Answers

Answer:

x=-7/6, y=-8/3. (-7/6, -8/3).

Step-by-step explanation:

6x-3y=1

y=-2x-5

-------------

6x-3(-2x-5)=1

6x+6x+15=1

12x+15=1

12x=1-15

12x=-14

x=-14/12=-7/6

y=-2(-7/6)-5=7/3-5=7/3-15/3=-8/3

A is the midpoint of segment bouded by (-2, 3) & (6, -1). B is a point at 3/4 of distance from (4, 3) to (0, -3). Find the equation of AB.

Answers

The equation of line segment AB, whose endpoints are A(x, y) = (2, 1) and B(x, y) = (1, - 1.5), is P(x, y) = (1 - k) · (2, 1) + k · (1, - 1.5), for 0 ≤ k ≤ 1.

How to derive the equation of a line segment

In this problem we need to determine the equation that described the line segment AB, whose endpoints are points within other two line segments. The location of a point can be found by means of the following formula:

P''(x, y) = P(x, y) + k · [P'(x, y) - P(x, y)], 0 ≤ k ≤ 1

Where:

P(x, y), P'(x, y) - Endpoints of line segment.k - Length ratio.P''(x, y) - Endpoint within line segment.

First, determine the locations of endpoints A and B:

A(x, y) = (- 2, 3) + 0.5 · [(6, - 1) - (- 2, 3)]

A(x, y) = (- 2, 3) + 0.5 · (8, - 4)

A(x, y) = (- 2, 3) + (4, - 2)

A(x, y) = (2, 1)

B(x, y) = (4, 3) + 0.75 · [(0, - 3) - (4, 3)]

B(x, y) = (4, 3) + 0.75 · (- 4, - 6)

B(x, y) = (4, 3) + (- 3, - 4.5)

B(x, y) = (1, - 1.5)

Second, derive the equation of line segment AB:

P(x, y) = A(x, y) + k · [B(x, y) - A(x, y)]

P(x, y) = (2, 1) + k · [(1, - 1.5) - (2, 1)]

P(x, y) = (1 - k) · (2, 1) + k · (1, - 1.5), for 0 ≤ k ≤ 1.

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Todd made $12,800 in interest by placing $40,000 in a savings account with simple interest for 4 years. What was the interest rate?

Answers

Answer: The interest rate is 0.16 or 16% ✅

Step-by-step explanation:

The interest rate is the percentage of the principal (the initial amount of money) that is paid as interest over a certain period of time. To find the interest rate in this scenario, we can use the formula:

Interest = Principal x Interest Rate x Time

In this case:

Interest = $12,800

Principal = $40,000

Time = 4 years

So we can set up the equation:

12800 = 40000 x Interest Rate x 4

To find the interest rate, we can divide both sides of the equation by 40000 x 4, which gives us:

12800 / (40000 x 4) = Interest Rate

We can simplify this fraction:

0.16 = Interest Rate

So the interest rate is 0.16 or 16%.

When Todd made $12,800 in interest by placing $40,000 in a savings account with simple interest for 4 years, the interest rate was 7.5%.

Simple interest is a formula for calculating the amount of interest that will be charged on a sum of money at a specific rate and during a specific time frame.

In contrast to compound interest, where the interest of one year's principal is added to the principal of the following year to calculate the interest, the principal sum in the case of simple interest stays the same.

Simple Interest = [tex]$ \frac{ \text P \times \text R \times \text T}{100}[/tex]

where,

P = Principal,

R = Rate of Interest in % per annum, and

T = Time, in years

Here given that,

Todd made $12,800 in interest, i.e. Simple Interest = 12,800

By placing $40,000 in a savings account, i.e. Principal = 40,000

with simple interest for 4 years, i.e. Time = 4 years

To find, interest rate

Substitute the data into the simple interest formula

[tex]$ \Rightarrow 12,800 = \frac{40,000 \times \text R \times 4}{100}[/tex]

[tex]$ \Rightarrow \text R = \frac{ 12,000 \times 100}{40,000 \times 4}[/tex]

[tex]$ \Rightarrow \text R = \frac{1,200,000}{160,000}[/tex]

⇒ R = 7.5%

Thus, the interest rate was 7.5%.

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What is the perimeter of trapezoid JKLM?
O√2+√5 units
O2+ √2+√5 units
O9+2√2 units
9+√2+√5 units

Answers

By applying pythagorean theorem and perimeter formula, The answer is D 9+[tex]\sqrt{2}[/tex] +[tex]\sqrt{5}[/tex]

What is pythagorean theorem?

The Pythagorean theorem is a fundamental result in mathematics named after the ancient Greek mathematician Pythagoras. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

What is perimeter?

Perimeter is a measure of the distance around the outside of a two-dimensional shape. It is the sum of the lengths of all the sides of the shape. The formula for perimeter depends on the shape. For example, the perimeter of a rectangle is twice the sum of its length and width, while the perimeter of a circle is equal to its circumference.

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The dimensions of the rectangular pool shown below are 60 yards by 30
yards. A fence will be built around the outside of the deck. The ratio of the
dimensions of the pool to the dimensions of the fence is How many yards
of fence should be purchased to enclose the deck?
60 Yards
30 Yards
A. 270 yards
B. 540 yards
C. 200 yards
D. 312 yards

Answers

540 yards should be purchased to enclose the deck and option B is correct.

What is rectangle?

A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. You might also describe it as follows: a quadrilateral that is equiangular, which indicates that all of its angles are equal. The parallelogram might also have a straight angle. Squares are rectangles with four equally sized sides.

A quadrilateral of the shape of a rectangle has four 90-degree vertices and equal parallel sides. As a result, it is sometimes referred to as an equirectangular rectangle. Because its opposite sides are equal and parallel, a rectangle is also known as a parallelogram.

Pools dimension = 1:3

Length = 60 × 3

= 180

Breadth= 30 × 4

= 90

Perimeter of rectangle = 2(l+b) = 2(90+180)

P = 2(270) = 540 yards

Thus, 540 yards should be purchased to enclose the deck and option B is correct

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Multiply. Write your answer as a decimal.
-0.05 × (-0.5)

Answers

The solution for the given decimal multiplication is 0.025.

What are decimal numbers and how do you multiply them?

To multiply two decimal numbers, first multiply the values without the decimal points.

The digits in the factors that follow the decimal point should now be added together.

Following that, the dot should be placed in the product so that it has the same number of digits as the total number of factors.

If there are more decimal places than digits in the number, place zeroes immediately after the dot.

If there are more decimal places present compared to digits in the product, fill the decimal places with zeroes immediately after the dot.

For the given question, multiply -5 * -5 = 25

Now the total digits after the decimal point in factors are 3.

But there are only 2 digits in 25.

So fill the decimal place with a zero.

Therefore the product of given decimal numbers is 0.025.

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What is 4, 700, 000 in scientific notation?
A. 4.7 x 105
B. 4.7 x 106
C. 4.7 x 107
D. 4.7 x 108

Answers

It would be 4.7 x 10^6
You would move the decimal place 6 numbers to the left.

Answer:

4,700,000 can be written as 4.7 × 10⁶ in scientific notation

Scientific notation:Scientific notation, also called power of 10 notation.It is a method of writing extremely large and small numbers.

For Example:

10 = 1 × 10¹ 100 = 1 × 10² = 10²1200 = 1.2 × 10³

Here,

1200 can also be written as

12 × 100

we know that, 100 = 10²

so,

12 × 10²

Remember

Decrease the base increase the power and vice-a-versa

1.2 × 10³

You put $600 in an account. The account earns $31 simple interest in 10 months. What is the annual interest rate? need % answer

Answers

The annual interest rate that yields $31 from $600 in 10 months is 6.2%

What is an interest earned on an amount?

An interest is the amount earned by placing a sum of money in a savings account or in an investment.

The principal amount you put in the account, P = $600

The amount (interest) earned in 6 months, I = $31

The duration over which the interest of $31 is earned, T = 10 months

The annual interest rate is found as follows;

The annual interest rate is the interest rate per year

The number of months in a year = 12 months

The formula for the simple interest, I, is presented as follows;

I = P·R·T

Where;

I = The simple interest earned

P = The principal amount saved in the account

R = The annual interest rate

T = The time duration the amount was saved in years

12 months = 1 year

Therefore; 10 months = (1/12) × 10 year = (5/6) year

The time, T, of 10 months = (5/6) year

31 = 600 × R × (5/6)

R = 31/(600 × (5/6)) = 0.062

0.062 in percentage is 0.062 × 100 = 6.2%

The annual interest rate, R = 6.2%

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What is the properties of (10c - 4) + 11d?

Answers

Answer:

10c-4+11d Welcome :D

Step-by-step explanation:

Jorge's trip takes the same amount of time as Zahara's but he traveled a shorter distance. Does Jorge drive faster or slower Zahara's dad

Answers

Answer: Slower

Step-by-step explanation:

Speed is a measure of how quickly an object moves, and it is calculated as distance traveled divided by time. Since Jorge and Zahara's dad traveled the same amount of time but Jorge traveled a shorter distance, it can be inferred that Zahara's dad traveled at a faster speed.

Another way to understand this is that, if Jorge and Zahara's dad were driving at the same speed, they would have traveled the same distance in the same amount of time. However, since Jorge traveled a shorter distance than Zahara's dad, it can be inferred that Jorge was driving at a slower speed.

Answer:

Step-by-step explanation:

3 / 3/4=, 5 / 3/4=, 8 / 4/7=, 6 / 3/5=, 2 / 5/8=, and 4 / 8/9= simplify them all

Answers

On simplifying the values obtained for the fractions are - 3 / 3/4= 4, 5 / 3/4= 6.66, 8 / 4/7= 14, 6 / 3/5= 10, 2 / 5/8= 3.2 and 4 / 8/9= 4.5.

What is fraction?

In mathematics, a fraction is represented by a number that identifies a portion of a whole. A fraction is a component or section taken from a whole, which can be any number, a certain amount, or an object.

The first fraction is 3/(3/4).