From the image, I can see the problem involves determining several values for a garden, which includes sections for Carrots, Beets, and Turnips.

Let's break down the questions:

The garden diagram:

• Carrots: 15 ft by 8 ft
• Beets: 6 ft by 12 ft
• Turnips: 12 ft by 11 ft

Questions:

1. What is the area of the entire garden? To find the total area, sum the areas of each section: $\text{Area of Carrots} = 15 \, \text{ft} \times 8 \, \text{ft} = 120 \, \text{sq ft}$ $\text{Area of Beets} = 6 \, \text{ft} \times 12 \, \text{ft} = 72 \, \text{sq ft}$ $\text{Area of Turnips} = 12 \, \text{ft} \times 11 \, \text{ft} = 132 \, \text{sq ft}$ $\text{Total area of the garden} = 120 + 72 + 132 = 324 \, \text{sq ft}$

2. What is the area of the beet patch? Already calculated above: $\text{Area of Beets} = 72 \, \text{sq ft}$

3. How much larger is the carrot patch than the beet patch? $\text{Difference} = 120 \, \text{sq ft} - 72 \, \text{sq ft} = 48 \, \text{sq ft}$

4. What is the smallest patch? What is the size of that patch? The smallest patch is the Beet patch, with an area of 72 sq ft.

5. What is the perimeter of the entire garden? The perimeter calculation requires combining all the outer lengths. Based on the diagram:

   • The garden forms a rectangle of dimensions 27 ft (length) by 23 ft (width). So, the perimeter is: $\text{Perimeter} = 2 \times (27 + 23) = 2 \times 50 = 100 \, \text{ft}$

Challenge:

Mrs. Jones wants to change the garden so the carrot patch is exactly 50% of the entire garden area.

Right now, the carrot patch is 120 sq ft, and the total garden area is 324 sq ft. She needs to make the carrot patch: $50\% \times 324 \, \text{sq ft} = 162 \, \text{sq ft}$ Thus, the carrot patch needs to increase its area by: $162 \, \text{sq ft} - 120 \, \text{sq ft} = 42 \, \text{sq ft}$ To achieve this, she would need to alter the dimensions of the carrot patch.

Do you want further clarification or details on any of the answers? Here are some related questions to deepen your understanding:

1. What would the dimensions of the carrot patch be to make its area exactly 162 sq ft?
2. How could Mrs. Jones alter the layout to keep the total perimeter the same while increasing the carrot patch size?
3. If Mrs. Jones wanted the turnip patch to be reduced by 10%, what would the new dimensions be?
4. How would you calculate the cost of fencing the entire garden if the cost per foot is $5?
5. What would happen to the perimeter if Mrs. Jones rearranged the garden into a square?

Tip: When dealing with composite shapes, always calculate the total area and perimeter by breaking it down into simpler shapes like rectangles.