Question

this is a multi - part item. pam’s eye - level height is 256 feet above sea level and adam’s eye - level height is 400 feet above sea level. what expression shows how much farther adam can see to the horizon? use the formula $d=sqrt{\frac{3h}{2}},hgeq0$, with $d$ being the distance they can see in miles and $h$ being their eye - level height in feet. in simplest terms, adam can see $asqrt{b}$ feet farther than pam. $a = square$, $b=square$
$sqrt{\frac{3(400)}{2}}-sqrt{\frac{3(256)}{2}}$
$sqrt{\frac{3(256)}{2}}-sqrt{\frac{3(400)}{2}}$
$sqrt{\frac{3(400)}{2}}+sqrt{\frac{3(256)}{2}}$

Explanation:

Step1: Simplify Adam's distance

Adam's height \( h = 400 \), so his distance \( d_{Adam} = \sqrt{\frac{3(400)}{2}} = \sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6} \) (wait, no, wait: \( \frac{3\times400}{2}=600 \), but let's do it step by step. Wait, the formula is \( d = \sqrt{\frac{3h}{2}} \). So for Adam: \( \frac{3\times400}{2} = 600 \), so \( d_{Adam} = \sqrt{600} \). For Pam: \( h = 256 \), so \( \frac{3\times256}{2} = 384 \), so \( d_{Pam} = \sqrt{384} \). Now, the difference is \( \sqrt{600} - \sqrt{384} \). Let's simplify both radicals.

Simplify \( \sqrt{600} \): \( 600 = 100 \times 6 \), so \( \sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6} \). Wait, no: \( 3\times400/2 = 600 \), so \( \sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6} \)? Wait, no, 3400 is 1200, divided by 2 is 600. So \( \sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6} \). Wait, but 3256 is 768, divided by 2 is 384. So \( \sqrt{384} = \sqrt{64 \times 6} = 8\sqrt{6} \). Oh! Wait, 384 = 64 6? 646=384? 646=384, yes. And 600=1006? 100*6=600, yes. So then the difference is \( 10\sqrt{6} - 8\sqrt{6} = (10 - 8)\sqrt{6} = 2\sqrt{6} \). Wait, but that would mean a=2, b=6? Wait, let's check again.

Wait, the formula is \( d = \sqrt{\frac{3h}{2}} \). So for Adam: \( h = 400 \), so \( \frac{3*400}{2} = 600 \), so \( d_{Adam} = \sqrt{600} \). For Pam: \( h = 256 \), so \( \frac{3*256}{2} = 384 \), so \( d_{Pam} = \sqrt{384} \). Now, let's factor 600 and 384 to see if they have a common factor. 600 = 100 6, 384 = 64 6. So \( \sqrt{600} = \sqrt{100*6} = 10\sqrt{6} \), \( \sqrt{384} = \sqrt{64*6} = 8\sqrt{6} \). Then the difference is \( 10\sqrt{6} - 8\sqrt{6} = 2\sqrt{6} \). So a=2, b=6. Wait, but let's check the initial expression. The expression given is \( \sqrt{\frac{3(400)}{2}} - \sqrt{\frac{3(256)}{2}} \), which is \( \sqrt{600} - \sqrt{384} \). Simplifying each:

\( \sqrt{600} = \sqrt{100 \times 6} = 10\sqrt{6} \)

\( \sqrt{384} = \sqrt{64 \times 6} = 8\sqrt{6} \)

Subtracting them: \( 10\sqrt{6} - 8\sqrt{6} = 2\sqrt{6} \). So in the form \( a\sqrt{b} \), a=2 and b=6.

Step2: Verify the simplification

Check the factorization of 600: 600 ÷ 100 = 6, so \( \sqrt{600} = 10\sqrt{6} \). 384 ÷ 64 = 6, so \( \sqrt{384} = 8\sqrt{6} \). Subtracting gives \( (10 - 8)\sqrt{6} = 2\sqrt{6} \), so a=2, b=6.

Answer:

a = 2, b = 6