Problem 56 - Algebra 2

Question

ACT/SAT Which of the following is closest to \(\sqrt[3]{7.32} ?\) A 1.8 B 1.9 C 2.0 D 2.1

Step-by-Step Solution

Answer

The closest value is 2.0.

Step 1: Understanding the Problem

We need to estimate the cube root of 7.32 and choose the option that is closest to this value.

Step 2: Estimate Initial Values

To have an idea of where the cube root lies, consider the perfect cubes around 7.32. We know that \(2^3 = 8\) and \(1.9^3\approx 6.859\). This means the cube root of 7.32 is between 1.9 and 2.0.

Step 3: Calculate Precise Values

Calculate \(1.9^3\) and \(2.0^3\) to see which one is closer to 7.32. \(1.9^3 = 1.9 \times 1.9 \times 1.9 = 3.61 \times 1.9 = 6.859\); \(2.0^3 = 2 \times 2 \times 2 = 8\).

Step 4: Analyze the Calculations

The number 7.32 is closer to 8 than to 6.859, suggesting that 2.0 may be the closer approximation.

Step 5: Refine the Estimate

We compare the error values to determine precision. The difference between 8 and 7.32 is smaller than the difference between 7.32 and 6.859. Hence, 2.0 is closer.

Key Concepts

Estimation

When we estimate the cube root of a number, especially in cases where we need a quick approximation, we're looking for a simple and speedy comparison.

The first step is to identify perfect cubes near the given number. This hints at where possible cube root values may lie. For example, with 7.32, we check around known values such as 1.9 and 2.0.

These known values give us benchmarks to understand if our target number is closer to one or the other. Estimation doesn’t require precise calculations but instead relies on identifying a range in which the number likely falls.

Perfect Cubes

Perfect cubes are numbers like 1, 8, 27, etc., where a whole number multiplied by itself three times gives these results. These are useful for identifying potential cube roots quickly.

In our exercise, we identified two perfect cubes of interest: the cube of 2, which is 8, and 1.9, which approximates to 6.859 when cubed.

Knowing perfect cubes around our target helps simplify the step to pin down the accurate cube root. This makes perfect cubes invaluable in mental math and quick assessments.

Error Analysis

Error analysis in estimation involves comparing how far each approximation is from the actual value.

You'll subtract the estimated cubed numbers from the actual number, in this case, 7.32.

• The difference between 8 and 7.32 is small.
• In contrast, the gap between 7.32 and 6.859 is larger.

By understanding and comparing these differences, we can determine which estimate is more accurate.

This methodology allows us to refine estimates, ensuring we make the best decision with given options.