Two angles supplementary to the same angle must be congruent.

Consider the parabola r(t)equalsleft angle at squared plus 1 comma t right angle, for minusinfinityless thantless thaninfinity

kodGreya [7K]

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;

$r{}'(t)=$

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 ,

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

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

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

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

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>

6 0

3 years ago

A high school band asked a number of students whether they would like blue, gold, or both for the uniforms for the band. The res

Anna35 [415]

Given a Venn diagram showing the number of students that like blue uniform only as 32, the number of students that like gold uniform only as 25, the number of students that like blue and gold uniforms as 12 and the number of students that like neither blue nor gold uniform as 6.

Thus, the total number of students interviewed is 75.

Recall that relative frequency of an event is the outcome of the event divided by the total possible outcome of the experiment.

From the relative frequency table, a represent the relative frequency of the students that like gold but not blue.
From the Venn, diagram, the number of students that like gold uniform only as 25, thus the relative frequency of the students that like gold but not blue is given by
$\frac{25}{75} = \frac{1}{3}=0.33=33\%$

Therefore, a = 33% to the nearest percent.

Similarly, from the relative frequency table, b represent the relative frequency of the students that like blue but not gold.
From the Venn, diagram, the number of students that like blue uniform only as 32, thus the relative frequency of the students that like gold but not blue is given by
$\frac{32}{75} = 0.427=42.7\%$

Therefore, b = 43% to the nearest percent.

7 0

3 years ago

A rectangular prism has volume 12x3 − 11x2 − 5x. Which expressions can represent the dimensions of the prism?

Nonamiya [84]

The expressions that can represent the dimensions of the prim is (x), (3x+1) and (4x-1).

3 0

3 years ago

How do you find inverses

MrMuchimi

How To Find Inverses:

1. First, replace f(x) with y . ...

2. Replace every x with a y and replace every y with an x .