Respuesta :
Answer:
The plate's area is increasing at the rate of 10.81 cm²/min when the radius is 43 cm.
Step-by-step explanation:
The area of a circle is given by the following formula:
[tex]A = \pi r^{2}[/tex]
In which the area is measured in cm².
Its radius increases at a rate of 0.04 cm divided by min.
This means that [tex]\frac{dr}{dt} = 0.04[/tex]
At what rate is the plate's area increasing when the radius is 43 cm?
This is [tex]\frac{dA}{dt}[/tex] when [tex]r = 43[/tex]
[tex]A = \pi r^{2}[/tex]
Applying implicit differentitation
We have two variables(A and r), so
[tex]\frac{dA}{dt} = 2r\pi \frac{dr}{dt}[/tex]
[tex]\frac{dA}{dt} = 2*43\pi*0.04[/tex]
[tex]\frac{dA}{dt} = 10.81[/tex]
The plate's area is increasing at the rate of 10.81 cm²/min when the radius is 43 cm.
This question is based on implicit differentiation.Therefore, the plate's area increasing when the radius is 43 cm with the rate of 10.81[tex]cm^2/min[/tex].
Given:
When a circular plate of metal is heated, its radius increases at a rate of 0.04 cm divided by min.
According to the question,
As we know that, the area of a circle is given by the following formula:
[tex]A = \pi r^2[/tex]
It is given that, its radius increases at a rate of 0.04 cm divided by min.
[tex]i.e. \, \dfrac{dr}{dt} = 0.04\\[/tex]
Now calculating the rate at which plate's area increasing when the radius is 43 cm,
[tex]i.e. \dfrac{dA}{dt} \, when \, r = 43 cm[/tex]
By applying implicit differentiation,
[tex]\dfrac{dA}{dt} = 2r\pi \dfrac{dr}{dt} \\\\\dfrac{dA}{dt} = 2 \times 43\pi \, 0.04\\\\\dfrac{dA}{dt} = 10.81 \, cm^2/min[/tex]
Therefore, the plate's area increasing when the radius is 43 cm with the rate of 10.81[tex]cm^2/min[/tex].
For more details, prefer this link:
https://brainly.com/question/24220930
