When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.02 cm divided by min0.02 cm/min. at what rate is the plate's area increasing when the radius is 4040 cm?

Since the plate is circular, therefore the area of the plate is jut equal to the area of a circle, so:

Area of plate = πr² = A

Taking the derivative:

dA / dr = 2πr                                                                           ---> 1

By the idea of partial differentiation, the equation can also take in the form of:
dA/dt = dA/dr x dr/dt                                                             ---> 2

Where we are given that:
change in radius over time = dr/dt = 0.02  cm/min
change in area with changing radius = dA/dr = 2πr              ---> from equation 1
at r = 40 
dA/dr = 2π(40) = 80π 

Substituting all the known values into equation 2:
dA/dt = (80π)(0.02)

dA/dt = 1.6π cm^2 /s = 5.03 cm^2/s

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this is the answer for you hope it

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.