Part a: Assume that the height of your cylinder is 8 inches. Think of A as a function of r, which we may express as&nbsp;A(r)=2πr2+16πr. What is the domain of&nbsp;A(r)? In other words, for which values of r is&nbsp;A(r)&nbsp;defined?&nbsp;Part b: Find the inverse function to&nbsp;A(r). Your answer should look like&nbsp;r=&nbsp;"some expression involving A".&nbsp;r(A)=&nbsp;Hints:&nbsp;1) You must find r to calculate an inverse function.2)Here you could start with&nbsp;A=2πr2+16πr. This equation is the same as&nbsp;A=2πr2+16πr−A=0. Do you recognize this as a quadratic equation&nbsp;ax2+bx+c=0&nbsp;where the variable x is r? The coefficients would be&nbsp;2π&nbsp;for a,&nbsp;16π&nbsp;for b, and&nbsp;−A&nbsp;for c.&nbsp;3)You can solve for r using the quadratic formula even though the constant term c is a symbol here.&nbsp;Part c: If the surface area is 225 square inches, then what is the radius r? In other words, evaluate&nbsp;r(225). Round your answer to 2 decimal places.&nbsp;Need Part A

Question

Part a: Assume that the height of your cylinder is 8 inches. Think of A as a function of r, which we may express as&amp;nbsp;A(r)=2πr2+16πr. What is the domain of&amp;nbsp;A(r)? In other words, for which values of r is&amp;nbsp;A(r)&amp;nbsp;defined?&amp;nbsp;Part b: Find the inverse function to&amp;nbsp;A(r). Your answer should look like&amp;nbsp;r=&amp;nbsp;&quot;some expression involving A&quot;.&amp;nbsp;r(A)=&amp;nbsp;Hints:&amp;nbsp;1) You must find r to calculate an inverse function.2)Here you could start with&amp;nbsp;A=2πr2+16πr. This equation is the same as&amp;nbsp;A=2πr2+16πr−A=0. Do you recognize this as a quadratic equation&amp;nbsp;ax2+bx+c=0&amp;nbsp;where the variable x is r? The coefficients would be&amp;nbsp;2π&amp;nbsp;for a,&amp;nbsp;16π&amp;nbsp;for b, and&amp;nbsp;−A&amp;nbsp;for c.&amp;nbsp;3)You can solve for r using the quadratic formula even though the constant term c is a symbol here.&amp;nbsp;Part c: If the surface area is 225 square inches, then what is the radius r? In other words, evaluate&amp;nbsp;r(225). Round your answer to 2 decimal places.&amp;nbsp;Need Part A

liannemdh · Accepted Answer

As only part A has been asked,(a)Assume that your cylinder is 8 inches tall. Consider A as a function of r, so we can write that as&amp;nbsp;A(r)=2πr2+16πr&amp;nbsp;.What is the domain of&amp;nbsp;A(r)? In other words, for which values of r is&amp;nbsp;A(r)&amp;nbsp;defined?A(r)=2πr2+16πr&amp;nbsp;...........(1)we have to write (1) in vertex form&amp;nbsp;A(r)=a(r–h)2+kA(r)=2π(r2+8r)=2π(r2–2×(−4r)+42–42)=2π{(r−(−4))2–16}=2π(r–(−4))2–32πcomparing this with the vertex form,&amp;nbsp;h=−4,&amp;nbsp;k=−32π, this is the minimum point of the quadratic equation (1)Hence range of&amp;nbsp;A(r)&amp;nbsp;is&amp;nbsp;[−32π,∞)The domain of&amp;nbsp;A(r)&amp;nbsp;is&amp;nbsp;(∞,∞).

madeleinejames20 · Accepted Answer

To find the domain of the function A(r)=2πr2+16πr, we need to determine the values of r for which A(r) is defined. In this case, we are assuming the height of the cylinder is 8 inches.The domain of A(r) consists of all possible values of r that make the expression A(r) well-defined.The first term in the expression is 2πr2, which is defined for all real values of r. Since squaring a real number always yields a real number, there are no restrictions on the values of r for this term.The second term in the expression is 16πr, which is also defined for all real values of r. Multiplying a real number by π or multiplying it by a constant does not introduce any restrictions on the domain.Therefore, the domain of A(r) is the set of all real numbers, or in mathematical notation:Domain&amp;nbsp;of&amp;nbsp;A(r)=ℝIn conclusion, the function A(r) is defined for all values of r in the set of real numbers.

Don Sumner · Accepted Answer

The given function A(r) involves the radius r of a cylinder and calculates the surface area A. In this case, the height of the cylinder is fixed at 8 inches.The domain of A(r) represents the set of valid values that r can take to produce a meaningful and defined result for A(r).For A(r) to be defined, both terms in the function must be defined. Let&#039;s analyze each term separately.Term 1: 2πr2The term 2πr2 represents the lateral surface area of the cylinder, which is defined for any non-negative real value of r. Since the radius cannot be negative, the domain for this term is r≥0.Term 2: 16πrThe term 16πr represents the area of the two bases of the cylinder, which is defined for any non-negative real value of r. Again, the radius cannot be negative, so the domain for this term is r≥0.To determine the overall domain of A(r), we need to consider the intersection of the domains of the individual terms. Since both terms have the same domain of r≥0, the domain of A(r) is also r≥0.Therefore, the domain of the function A(r), expressed in LaTeX markup, is [0,∞) or r∈[0,∞).

Eliza Beth13 · Accepted Answer

Result:A(r) is (−∞,∞)Solution:Since A(r) involves the terms r2 and r, we must ensure that these terms are valid for any real number input of r.1. For r2 to be defined, r must be a real number. This means that r can take any value from negative infinity to positive infinity, represented as (−∞,∞).2. For 16πr to be defined, r must also be a real number. Therefore, the domain for this term is the same as in step 1, (−∞,∞).3. Combining the two terms, 2πr2 and 16πr, we can say that the domain of A(r) is the intersection of the domains of these two terms. In other words, the domain is the set of all values of r for which both terms are defined.Since both terms are defined for (−∞,∞), the domain of A(r) is also (−∞,∞).Therefore, the domain of A(r) is (−∞,∞).

Part a: Assume that the height of your cylinder is 8 inches. Consider A as a fun

Answered question

Answer & Explanation

New Questions in Algebra I