Math Problem Statement

b is directly proportional to the square root of a. Complete the table.

Solution

The problem in the image states that bb is directly proportional to the square root of aa, and asks you to complete the table.

We can express the relationship mathematically as: b=kab = k \sqrt{a} where kk is the constant of proportionality.

Step 1: Find kk

Using the values a=1a = 1 and b=9b = 9 from the table, we substitute these into the equation: 9=k19 = k \sqrt{1} Since 1=1\sqrt{1} = 1, we find: k=9k = 9

Step 2: Use kk to complete the table

Now that we know k=9k = 9, we can use this to find the missing values for bb and the corresponding aa.

  • When a=36a = 36: b=9×36=9×6=54b = 9 \times \sqrt{36} = 9 \times 6 = 54

So the missing value for bb when a=36a = 36 is 54.

Thus, the completed table looks like this:

| aa | 1 | 36 |
| bb | 9 | 54 |

Would you like any further clarification on this solution? Here are five related questions you might find useful:

  1. What if bb were inversely proportional to the square root of aa?
  2. How would the table change if bb were proportional to a2a^2 instead of a\sqrt{a}?
  3. Can you graph the relationship between aa and bb for this proportional relationship?
  4. What are other examples of directly proportional relationships in mathematics?
  5. How do you determine the constant of proportionality in any relationship?

Tip: Always ensure to substitute known values into the proportionality equation to find the constant before using it for other calculations.

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Math Problem Analysis

Mathematical Concepts

Direct Proportion
Square Roots

Formulas

b = k * sqrt(a)

Theorems

Direct Proportionality

Suitable Grade Level

Grades 8-10