Question

question 5.
naomi is building a circular coffee table for her living room. she needs to find the area of the top of the table that has radius of ( 5 x ^ { \frac { 9 } { 2 } } y ^ { \frac { 2 } { 3 } } ) inches. the area of a circle can be found using ( a = pi r ^ { 2 } ).
what is the area top of the coffee table?
a. ( 25 pi x ^ { 9 } y ^ { \frac { 4 } { 3 } } )
b. ( 10 pi x ^ { 18 } y ^ { 8 } )
c. ( 15 pi x ^ { 9 } y ^ { 4 } )
d. ( 15 pi x ^ { 9 } y ^ { \frac { 2 } { 3 } } )

Explanation:

Step1: Substitute the radius into the area formula

Given \(r = 5x^{\frac{3}{2}}y^{\frac{2}{3}}\), and \(A=\pi r^{2}\). Then \(A=\pi(5x^{\frac{3}{2}}y^{\frac{2}{3}})^{2}\).

Step2: Apply the power - of - a - product rule \((ab)^{n}=a^{n}b^{n}\)

\(A=\pi\times5^{2}\times(x^{\frac{3}{2}})^{2}\times(y^{\frac{2}{3}})^{2}\).
Since \(5^{2}=25\), \((x^{\frac{3}{2}})^{2}=x^{\frac{3}{2}\times2}=x^{3}\), \((y^{\frac{2}{3}})^{2}=y^{\frac{2}{3}\times2}=y^{\frac{4}{3}}\).

Step3: Calculate the final result

\(A = 25\pi x^{3\times3}y^{\frac{4}{3}}\) (using \((a^{m})^{n}=a^{mn}\) again for \(x\) part: \((x^{3})^{3}=x^{9}\)).

Answer:

A. \(25\pi x^{9}y^{\frac{4}{3}}\)