Chapter 2.6, Problem 1AYU

1. $P\text{ = }\left(x,y\right)$ be a point on the graph of $y\text{ = }{x}^2-\text{ 8}$.

(a) Express the distance d from P to the origin as a function of x.

(b) What is d if $x\text{ = 0}$?

(c) What is d if $x\text{ = 1}$?

(d) Use a graphing utility to graph $d = d\left(x\right)$.

(e) For what values of x is d smallest?

To determine

To find:

a. To express the distance $d$ from $p$ to the origin as a function of $x$.

Answer to Problem 1AYU

Solution:

a. $\sqrt{x^4-15x^2+64}$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

a. To express the distance $d$ from $P$ to the origin as a function of $x$:

$d=\sqrt{{\left(x-0\right)}^2+{\left(y-0\right)}^2}=\sqrt{x^2+y^2}$

Since $P$ is a point on the graph $y=x^2-8$. Substitute $x^2-8$ for $y$. Then

$d\left(x\right)=\sqrt{x^2+{\left(x^2-8\right)}^2}$

$=\sqrt{x^2+x^4-16x^2+64}$

$=\sqrt{x^4-15x^2+64}$

To determine

To find:

b. To find $d$ if $x=0$.

Answer to Problem 1AYU

Solution:

b. 8

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

b. To find $d$ if $x=0$:

$d\left(0\right)=\sqrt{64}=8$

To determine

To find:

c. To find $d$ if $x=1$.

Answer to Problem 1AYU

Solution:

c. $7.07$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

c. To find $d$ if $x=1$:

$d\left(1\right)=\sqrt{65-15}=\sqrt{50}=7.07$

To determine

To find:

d. To graph $d=d\left(x\right)$.

Answer to Problem 1AYU

Solution:

d. $x\approx -x2.74 \text{or} 2.74$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

d. To graph $d=d(x)$:

To determine

To find:

e. To find values of $x$ is $d$ smallest.

Answer to Problem 1AYU

Solution:

e. $x\approx -2.74\text{or}2.74$

Explanation of Solution

Given:

$y=x^2-8$

Calculation:

e. To find values of $x$ is $d$ smallest:

From the graph it can be seen that when $x=-2.74 \text{or} 2.74, d\left(=2.78\right)$ attains its minimum.

Therefore, $d$ is smallest when $x\approx -2.74 \text{or} 2.74$.