The angular velocity of an object is changed according to the law: w=A+Bt^2, where A=2 rad/s^2 , B=3 rad/s^3. For what angle will the object have rotated from time t1=1s to time t2=3s?
Answers
Answer:
We have been asked to calculate the following expression:
\longrightarrow \Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟶[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
Hint: Try to simplify the expression.
Expression - A mathematical symbol, or combination of symbols, represent a value, or relation. Example: 2 + 2 = 42+2=4 .
Calculation:
Let's start solving our problem and understanding every step to achieve our end result.
\Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
Step 1. Calculating the power of the power.
\implies \bigg(\dfrac{2}{3}\bigg)^{6} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹(
3
2
)
6
×(
3
1
)
−4
×3
−4
×6
−1
Step 2. When raising a fraction to the power, raise the numerator and denominator each to the power.
\implies \dfrac{2^{6}}{3^{6}} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹
3
6
2
6
×(
3
1
)
−4
×3
−4
×6
−1
Step 3. Calculate the power.
\implies \dfrac{64}{729} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹
729
64
×(
3
1
)
−4
×3
−4
×6
−1
Step 4. When raising a fraction to the power, raise the numerator and denominator each to the power.
\implies \dfrac{64}{729} \times \dfrac{1^{-4}}{3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
3
−4
1
−4
×3
−4
×6
−1
Step 5. If the exponent is negative, change it to a fraction.
\implies \dfrac{64}{729} \times \dfrac{\frac{1}{1^{4}}}{3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
3
−4
1
4
1
×3
−4
×6
−1
Step 6. Arrange the expression of the complex fraction.
\implies \dfrac{64}{729} \times \dfrac{1}{1^{4} \times 3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
1
4
×3
−4
1
×3
−4
×6
−1
Step 7. Arrange the terms multiplied by fractions.
\implies \dfrac{64 \times 3^{-4} \times 6^{1}}{729(1^{4} \times 3^{-4})}⟹
729(1
4
×3
−4
)
64×3
−4
×6
1
Step 8. Calculate the fraction.
\implies \dfrac{64}{2 \times 3^{7}}⟹
2×3
7
64
Step 9. Reduce the fraction.
\implies \dfrac{32}{3^{7}}⟹
3
7
32
Step 10. Calculate the power.
\implies \dfrac{32}{2187}⟹
2187
32
Final answer:
\boxed{\Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1} = \dfrac{32}{2187}}
[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
=
2187
32
Explanation:
We have been asked to calculate the following expression:
\longrightarrow \Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟶[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
Hint: Try to simplify the expression.
Expression - A mathematical symbol, or combination of symbols, represent a value, or relation. Example: 2 + 2 = 42+2=4 .
Calculation:
Let's start solving our problem and understanding every step to achieve our end result.
\Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
Step 1. Calculating the power of the power.
\implies \bigg(\dfrac{2}{3}\bigg)^{6} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹(
3
2
)
6
×(
3
1
)
−4
×3
−4
×6
−1
Step 2. When raising a fraction to the power, raise the numerator and denominator each to the power.
\implies \dfrac{2^{6}}{3^{6}} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹
3
6
2
6
×(
3
1
)
−4
×3
−4
×6
−1
Step 3. Calculate the power.
\implies \dfrac{64}{729} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1}⟹
729
64
×(
3
1
)
−4
×3
−4
×6
−1
Step 4. When raising a fraction to the power, raise the numerator and denominator each to the power.
\implies \dfrac{64}{729} \times \dfrac{1^{-4}}{3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
3
−4
1
−4
×3
−4
×6
−1
Step 5. If the exponent is negative, change it to a fraction.
\implies \dfrac{64}{729} \times \dfrac{\frac{1}{1^{4}}}{3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
3
−4
1
4
1
×3
−4
×6
−1
Step 6. Arrange the expression of the complex fraction.
\implies \dfrac{64}{729} \times \dfrac{1}{1^{4} \times 3^{-4}} \times 3^{-4} \times 6^{-1}⟹
729
64
×
1
4
×3
−4
1
×3
−4
×6
−1
Step 7. Arrange the terms multiplied by fractions.
\implies \dfrac{64 \times 3^{-4} \times 6^{1}}{729(1^{4} \times 3^{-4})}⟹
729(1
4
×3
−4
)
64×3
−4
×6
1
Step 8. Calculate the fraction.
\implies \dfrac{64}{2 \times 3^{7}}⟹
2×3
7
64
Step 9. Reduce the fraction.
\implies \dfrac{32}{3^{7}}⟹
3
7
32
Step 10. Calculate the power.
\implies \dfrac{32}{2187}⟹
2187
32
Final answer:
\boxed{\Bigg[\bigg(\dfrac{2}{3}\bigg)^{2}\Bigg]^{3} \times \bigg(\dfrac{1}{3}\bigg)^{-4} \times 3^{-4} \times 6^{-1} = \dfrac{32}{2187}}
[(
3
2
)
2
]
3
×(
3
1
)
−4
×3
−4
×6
−1
=
2187
32