A sphere has a radius of 7𝑥 + 7. Which polynomial in standard form best describes the total surface area of the sphere? Use the formula 𝑆 = 4𝜋𝑟^2 for the surface area of a sphere.
a. 196𝜋𝑥^2 − 392𝜋𝑥 + 196𝜋
b. 196𝜋𝑥^2 + 392𝜋𝑥 + 196𝜋
c. 196𝜋𝑥^2 − 392𝜋𝑥 − 196𝜋
d. 196𝜋𝑥^2 − 3926𝜋𝑥 − 196𝜋
A = 4 π r²
A = 4 π ( 7 x + 7 )² =
4 π [ ( 7 x )² + 2 ∙ 7 x ∙ 7 + 7² ] =
4 π ( 49 x² + 98 x + 49 ) =
4 π ∙ 49 x² + 4 π ∙ 98 x + 4 π ∙ 49 =
196 π x² + 392 π x + 196 π
Answer b
Well, let's see. The formula for the surface area of a sphere is S = 4πr^2. Now, if the radius of the sphere is 7𝑥 + 7, then we can substitute that value into the formula.
So we have S = 4π(7𝑥 + 7)^2.
Expanding this out, we get S = 4π(49𝑥^2 + 98𝑥 + 49).
And simplifying further, we have S = 196π𝑥^2 + 392π𝑥 + 196π.
So the polynomial in standard form that best describes the total surface area of the sphere is 196π𝑥^2 + 392π𝑥 + 196π.
Therefore, the correct answer is (b) 196π𝑥^2 + 392π𝑥 + 196π.
And that's how we roll, or should I say, sphere!
To find the total surface area of a sphere, we can use the formula 𝑆 = 4𝜋𝑟^2, where 𝑆 represents the surface area and 𝑟 is the radius of the sphere.
Given that the radius of the sphere is 7𝑥 + 7, we can substitute this expression into the formula:
𝑆 = 4𝜋(7𝑥 + 7)^2
To simplify this expression, we need to expand (7𝑥 + 7)^2 using the binomial theorem:
(7𝑥 + 7)^2 = (7𝑥)^2 + 2(7𝑥)(7) + (7)^2
= 49𝑥^2 + 98𝑥 + 49
Substituting this back into the surface area formula, we have:
𝑆 = 4𝜋(49𝑥^2 + 98𝑥 + 49)
= 196𝜋𝑥^2 + 392𝜋𝑥 + 196𝜋
Therefore, the polynomial in standard form that best describes the total surface area of the sphere is:
b. 196𝜋𝑥^2 + 392𝜋𝑥 + 196𝜋
To find the total surface area of a sphere, we can use the formula 𝑆 = 4𝜋𝑟^2, where 𝑆 is the surface area and 𝑟 is the radius of the sphere.
In this case, the radius is given as 7𝑥 + 7. To find the surface area of the sphere, we need to square the radius and multiply it by 4𝜋.
Let's substitute the value of the radius into the formula and simplify it:
𝑆 = 4𝜋(7𝑥 + 7)^2
Expanding the expression inside the parentheses using the binomial square formula (a + b)^2 = a^2 + 2ab + b^2:
𝑆 = 4𝜋(49𝑥^2 + 2 * 7 * 7𝑥 + 7^2)
= 4𝜋(49𝑥^2 + 98𝑥 + 49)
= 4𝜋 * 49𝑥^2 + 4𝜋 * 98𝑥 + 4𝜋 * 49
= 196𝜋𝑥^2 + 392𝜋𝑥 + 196𝜋
Therefore, the polynomial in standard form that best describes the total surface area of the sphere is option b: 196𝜋𝑥^2 + 392𝜋𝑥 + 196𝜋.