The shape of a colony of bacteria on a Petri dish is circular. Find the approximate increase in its area if its radius increases from 40 mm to 49 mm.

The estimated change in area is nothing mm2.

A = πr^2

∆A ≈ dr ∆r = 2π*40*(49-40) = 720π

actual change is π(49^2 - 40^2) = 801π
not very close, since ∆r is so big.

Ah, the growth of a bacterial colony! So, we have a circular shape on our hands, with an initial radius of 40 mm and a final radius of 49 mm. Let's calculate the approximate increase in area, shall we?

We know that the formula for the area of a circle is A = πr^2, where A represents the area and r is the radius.

Initially, the area (A1) with a radius of 40 mm is given by A1 = π(40)^2.

At the end, when the radius is 49 mm, the area (A2) can be calculated as A2 = π(49)^2.

Now, to find the approximate change in area, we subtract the initial area from the final area: A2 - A1.

Using this formula, the estimated change in area is nothing mm²... Oh wait, that can't be right! Let me recalculate that for you.

Calculating... recalculating... yup, I made a clownish mistake earlier. The estimated change in area is actually approximately 17,798 mm².

Keep in mind that this is an approximation since we used the radius values, not the precise measurements. But hey, that should give you a good idea of the increase in the bacterial colony's area.

To find the approximate increase in the area of the bacterial colony, we can use the formula for the area of a circle:

Area = π * r^2

Given that the radius increases from 40 mm to 49 mm, we can calculate the change in area as follows:

Old Area = π * (40 mm)^2
New Area = π * (49 mm)^2

To find the approximate increase in area, we can subtract the old area from the new area:

Approximate Increase in Area = New Area - Old Area
≈ (π * (49 mm)^2) - (π * (40 mm)^2)
≈ π * (49^2 mm^2 - 40^2 mm^2)

Simplifying the expression within parentheses:

≈ π * (2401 mm^2 - 1600 mm^2)
≈ π * 801 mm^2

Therefore, the approximate increase in area is approximately 801 mm².

To find the approximate increase in the area of the bacteria colony, we need to calculate the difference between the areas when the radius is 40 mm and when it is 49 mm.

The area of a circle can be calculated using the formula: A = πr², where A represents the area and r represents the radius.

When the radius is 40 mm, the initial area will be:
A1 = π(40 mm)²

When the radius increases to 49 mm, the final area will be:
A2 = π(49 mm)²

To find the approximate increase in area, we subtract the initial area from the final area:
Approximate increase in area = A2 - A1

Let's calculate it step by step:

Initial area (A1):
A1 = π(40 mm)²

Calculating A1:
A1 = π(1600 mm²)
A1 ≈ 5026.548 mm² (using the value of π as approximately 3.14159)

Final area (A2):
A2 = π(49 mm)²

Calculating A2:
A2 = π(2401 mm²)
A2 ≈ 7539.816 mm² (using the value of π as approximately 3.14159)

Approximate increase in area:
Approximate increase in area = A2 - A1
Approximate increase in area ≈ 7539.816 mm² - 5026.548 mm²
Approximate increase in area ≈ 2513.268 mm²

Therefore, the approximate increase in area of the bacteria colony is 2513.268 mm².