In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle?
Juan can measure the diameter as a length anywhere between $20 - 20\cdot 0.2 = 16$ and $20 + 20\cdot 0.2 = 24$  cm. The actual area of the circle is $\pi (20/2)^2=100\pi$ square cm, but Juan can compute the area anywhere in the range $\pi (16/2)^2=64 \pi$ square cm to $\pi (24/2)^2=144 \pi$ square cm. Using the lower bound of the range, Juan's error is $(100\pi - 64\pi)/(100\pi)=36\%$. Using the upper bound of the range, Juan's error is $(144\pi - 100\pi)/(100\pi)=44\%$. Thus, the largest possible percent error is $\boxed{44}$ percent.
44
Level 5
Prealgebra
Problem: In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of a circle, the formula used is A = πr², where r is the radius. The radius is half of the diameter. Given that the actual diameter is 20 cm, the actual radius is 10 cm. Juan's measurement of the diameter can vary due to an error of up to 20%. 

Calculating the maximum and minimum possible diameters based on the error:
- Maximum diameter = 20 cm + 20% of 20 cm = 20 cm + 4 cm = 24 cm.
- Minimum diameter = 20 cm - 20% of 20 cm = 20 cm - 4 cm = 16 cm.

Next, we calculate the corresponding radii:
- Maximum radius = 24 cm / 2 = 12 cm.
- Minimum radius = 16 cm / 2 = 8 cm.

Now, we compute the areas based on these radii:
- Area with maximum radius = π(12 cm)² = 144π cm².
- Area with minimum radius = π(8 cm)² = 64π cm².
- Actual area with correct radius = π(10 cm)² = 100π cm².

Next, we determine the percent error in the computed area:
- Percent error for maximum area = [(144π - 100π) / 100π] × 100% = (44π / 100π) × 100% = 44%.
- Percent error for minimum area = [(100π - 64π) / 100π] × 100% = (36π / 100π) × 100% = 36%.

The largest possible percent error in Juan's computed area occurs with the maximum radius, which is 44%. Thus, the final answer is 44.<|END_OF_LATENT|> The final answer is $44$.