To solve the equation
[tex]\[ 1.5 \cdot (4x)^2 = 12 \][/tex]
we follow these steps:
1. Simplify the equation:
Divide both sides of the equation by 1.5 to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
(4x)^2 = \frac{12}{1.5}
\][/tex]
2. Compute the right side:
[tex]\[
\frac{12}{1.5} = 8
\][/tex]
So the equation now reads:
[tex]\[
(4x)^2 = 8
\][/tex]
3. Take the square root of both sides:
[tex]\[
4x = \sqrt{8}
\][/tex]
4. Isolate [tex]\(x\)[/tex]:
Divide both sides by 4:
[tex]\[
x = \frac{\sqrt{8}}{4}
\][/tex]
5. Simplify the expression:
Recognize that [tex]\(\sqrt{8}\)[/tex] can be simplified as [tex]\(\sqrt{4 \cdot 2} = 2\sqrt{2}\)[/tex]:
[tex]\[
x = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2}
\][/tex]
Using a calculator, the approximate value of [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is 0.7071067811865476.
6. Round to the nearest hundredth:
[tex]\[
x \approx 0.71
\][/tex]
The value of [tex]\(x\)[/tex] that satisfies the given equation, rounded to the nearest hundredth, is
[tex]\(\boxed{0.71}\)[/tex].
