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Consider the parabola r(t)equalsleft angle at squared plus 1 comma t right angle, for minusinfinityless thantless thaninfinity, where a is a positive real number. Find all points on the parabola at which r and bold r prime are orthogonal.

Answers

Answer 1

Answer:

Given:- $r(t)=< at^2+1,t>$ ; $-\infty < t< \infty$ , where a is any positive real number.

Consider the helix parabolic equation :

$r(t)=< at^2+1,t>$

now, take the derivatives we get;

$r{}'(t)=<2at,1>$

As, we know that two vectors are orthogonal if their dot product is zero.

Here, $r(t) and r{}'(t)$ are orthogonal i.e, $r\cdot r{}'=0$

Therefore, we have ,

$< at^2+1,t>\cdot < 2at,1>=0$

$< at^2+1,t>\cdot < 2at,1>=<at^2+1\cdot\left(2at\right), t\cdot \left(1)>$

$=2a^2t^3+2at+t$

$2a^2t^3+2at+t=0$

take t common in above equation we get,

$t\cdot \left (2a^2t^2+2a+1\right )=0$

⇒ $t=0$ or $2a^2t^2+2a+1=0$

To find the solution for t;

take $2a^2t^2+2a+1=0$

The number $D = b^2 -4ac$ determined from the coefficients of the equation $ax^2 + bx + c = 0.$

The determinant $D=0-4(2a^2)(2a+1)$ $=-8a^2\cdot(2a+1)$

Since, for any positive value of a determinant is negative.

Therefore, there is no solution.

The only solution, we have t=0.

Hence, we have only one points on the parabola $r(t)=< at^2+1,t>$ i.e <1,0>

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In a recent election Corrine Brown received 13,436 more votes than Bill Randall. If the total number of votes was 119,444 find the number of votes for each candidate.

Answers

Answer:

53151 for Bill Randall

66565 for Corrine Brown

Step-by-step explanation:

X + X+13414 = 119716

2x + 13414 = 119716

2x = 106302

x=53151

119716 - 53151 = 66565

Help me please thank you

Answers

Answer:

36 miles

Step-by-step explanation:

If she can drive 12 miles only using 1/3 of a gallon of gas, that means if she uses a full gallon (3/3) she would be driving 36 miles.

12/1 x 1/3

36/1 x 3/3

Multiply the numerators by 3.

Aiden attended school for 408 minutes, played basketball for 138 minutes, and worked on homework for 111 minutes. Estimate how much time he spent doing these activities by rounding to the nearest ten first and then adding the rounded numbers.

Answers

Answer:

well 660

Step-by-step explanation:

well frist we will have to subtract 408 with every other number

408-138=270

270-111=159

this is what he had left

but if it's the other way around 410+140+110=660

What is credit? A. an arrangement in which you receive money, goods, or services now in exchange for the promise of payment later
B. an arrangement in which you receive goods or services in exchange for other goods and services
C. an arrangement in which you receive money now and pay it back later with fees

Answers

the answer is C.an arrangement in which you receive money now and pay it back later with fees

Answer:

Step-by-step explanation:

Raj recorded the scores of six of his classmates in the table.Math Scores
98
76
100
88
82
70

What is the range of their test scores?

Answers

The answer is 30. Range: largest number - smallest number = range

The data set shows the number of practice free throws players in a basketball competition made and the number of free throws they made in a timed competition.Practice throws 312 614 2 710 8Free throws 10 22 9 35 4 11 28 20A. Use technology to find the equation AND coefficient of determination for each type of regression model. Use the number of practice throws for the input variable and the number of free throws for the output variable. Round all decimal values to three places. Equation of regression modelCoefficient of determinationLinear model Quadratic model Exponential model B. EXPLAIN, using the information you found above, which model best fits the data set. How did you come to your conclusion?Note: write exponents like this x^2 if you need to.

Answers

Let

x -----> number of practice throws

y -----> the number of free throws

Part A

Linear Model

Using a Linear Regression Calculator

we have

ŷ = 2.352X - 0.852

see the attached figure

Remember that

With linear regression, the coefficient of determination is equal to the square of the correlation between the x and y variables

the coefficient r=0.919

Quadratic model

Using a Quadratic regression Calculator

we have

y=0.073x^2+1.205x+2.555

correlation coefficient r=0.925 ------> strong correlation

see the attached figure

Exponential model

Using an Exponential Regression Calculator

we have

y=4.229(1.17)^x

Correlation:r=0.913

see the attached figure

Part B

The model that best fits the data is the Quadratic model because its value of r is greater than the linear model and greater than the exponential model

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Consider the parabola r​(t)equalsleft angle at squared plus 1 comma t right angle​, for minusinfinityless thantless thaninfinity​, where a is a positive real number. Find all points on the parabola at which r and bold r prime are orthogonal.

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Consider the parabola r(t)equalsleft angle at squared plus 1 comma t right angle, for minusinfinityless thantless thaninfinity, where a is a positive real number. Find all points on the parabola at which r and bold r prime are orthogonal.