Answer :
To find the value of [tex]\( x \)[/tex] such that the terms [tex]\( 48x^2 \)[/tex], [tex]\( 64x^3 \)[/tex], and [tex]\( 36x^2 \)[/tex] are in proportion, we need these expressions to satisfy the condition for proportion:
For three terms [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] to be in proportion, the ratio [tex]\( \frac{a}{b} \)[/tex] should equal [tex]\( \frac{b}{c} \)[/tex].
1. Set the given expressions:
- [tex]\( a = 48x^2 \)[/tex]
- [tex]\( b = 64x^3 \)[/tex]
- [tex]\( c = 36x^2 \)[/tex]
2. Set up the proportion using these terms:
[tex]\[
\frac{48x^2}{64x^3} = \frac{64x^3}{36x^2}
\][/tex]
3. Simplify each part of the proportion:
- The left side [tex]\(\frac{48x^2}{64x^3}\)[/tex] simplifies as follows:
[tex]\[
\frac{48}{64} \times \frac{x^2}{x^3} = \frac{3}{4} \times \frac{1}{x} = \frac{3}{4x}
\][/tex]
- The right side [tex]\(\frac{64x^3}{36x^2}\)[/tex] simplifies as follows:
[tex]\[
\frac{64}{36} \times \frac{x^3}{x^2} = \frac{16}{9} \times x = \frac{16x}{9}
\][/tex]
4. Set the simplified expressions equal:
[tex]\[
\frac{3}{4x} = \frac{16x}{9}
\][/tex]
5. Solve the equation for [tex]\( x \)[/tex]:
- Multiply both sides by [tex]\( 4x \times 9 \)[/tex] to eliminate fractions:
[tex]\[
3 \times 9 = 16x \times 4x
\][/tex]
[tex]\[
27 = 64x^2
\][/tex]
- Solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{27}{64}
\][/tex]
- Take the square root of both sides to find [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{27}{64}} = \pm \frac{\sqrt{27}}{8}
\][/tex]
- Simplifying [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{3 \times 3 \times 3} = 3\sqrt{3}
\][/tex]
Therefore,
[tex]\[
x = \pm \frac{3\sqrt{3}}{8}
\][/tex]
Hence, the possible values for [tex]\( x \)[/tex] are [tex]\(-\frac{3\sqrt{3}}{8}\)[/tex] and [tex]\(\frac{3\sqrt{3}}{8}\)[/tex].
For three terms [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] to be in proportion, the ratio [tex]\( \frac{a}{b} \)[/tex] should equal [tex]\( \frac{b}{c} \)[/tex].
1. Set the given expressions:
- [tex]\( a = 48x^2 \)[/tex]
- [tex]\( b = 64x^3 \)[/tex]
- [tex]\( c = 36x^2 \)[/tex]
2. Set up the proportion using these terms:
[tex]\[
\frac{48x^2}{64x^3} = \frac{64x^3}{36x^2}
\][/tex]
3. Simplify each part of the proportion:
- The left side [tex]\(\frac{48x^2}{64x^3}\)[/tex] simplifies as follows:
[tex]\[
\frac{48}{64} \times \frac{x^2}{x^3} = \frac{3}{4} \times \frac{1}{x} = \frac{3}{4x}
\][/tex]
- The right side [tex]\(\frac{64x^3}{36x^2}\)[/tex] simplifies as follows:
[tex]\[
\frac{64}{36} \times \frac{x^3}{x^2} = \frac{16}{9} \times x = \frac{16x}{9}
\][/tex]
4. Set the simplified expressions equal:
[tex]\[
\frac{3}{4x} = \frac{16x}{9}
\][/tex]
5. Solve the equation for [tex]\( x \)[/tex]:
- Multiply both sides by [tex]\( 4x \times 9 \)[/tex] to eliminate fractions:
[tex]\[
3 \times 9 = 16x \times 4x
\][/tex]
[tex]\[
27 = 64x^2
\][/tex]
- Solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{27}{64}
\][/tex]
- Take the square root of both sides to find [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{27}{64}} = \pm \frac{\sqrt{27}}{8}
\][/tex]
- Simplifying [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{3 \times 3 \times 3} = 3\sqrt{3}
\][/tex]
Therefore,
[tex]\[
x = \pm \frac{3\sqrt{3}}{8}
\][/tex]
Hence, the possible values for [tex]\( x \)[/tex] are [tex]\(-\frac{3\sqrt{3}}{8}\)[/tex] and [tex]\(\frac{3\sqrt{3}}{8}\)[/tex].
