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Question

Question
16. Peter is building a fenced in area for his dog. He has 250 feet (ft.) of fencing to create an area for his dog. He is building it next to his house so he will only need fencing for 3 sides of the rectangular area. The diagram shows the width, d, of the area Peter is building for his dog.

The function A(d)=d(250−2d) represents the area that will be fenced in for Peter's dog, in square feet.

Complete the statements.

(2 points)
A reasonable scale for the horizontal axis of the function A(d)=d(250−2d) is
10 ft
, A reasonable scale for the vertical axis of the function A(d)=d(250−2d) is
2000 sq ft

Answer 1

To complete the statements regarding the function \( A(d) = d(250 - 2d) \), we need to consider how to properly represent the width \( d \) and the area \( A(d) \).

1. Horizontal Axis (Width \( d \)):

   • In the context of the problem, \( d \) represents the width of the fenced area that Peter is building.
   • Since Peter has 250 feet of fencing and he is using only 3 sides, the maximum width \( d \) could logically be half of 250 ft (that would maximize area) leading to a maximum \( d \) of 125 ft.
   • A reasonable scale for the horizontal axis could therefore be increments of 10 ft, allowing for clear data presentation up to the maximum width.

2. Vertical Axis (Area \( A(d) \)):

   • The area function calculates the fenced area based on the width \( d \).
   • The area \( A(d) \) can be calculated as \( A(d) = d(250 - 2d) \), which is a downward-opening parabola.
   • To find the maximum possible area, we can look for the vertex of the parabola. The vertex will occur at \( d = \frac{b}{2a} \) or \( d = \frac{250}{4} = 62.5 \) ft, leading to a maximum area.
   • Substituting \( d = 62.5 \) into the area formula gives \( A(62.5) = 62.5(250 - 2(62.5)) = 62.5(125) = 7812.5 \) sq ft.
   • A reasonable scale for the vertical axis can be then set to 2000 sq ft to adequately capture the area up to this maximum.

So the completed statements are:

• A reasonable scale for the horizontal axis of the function \( A(d) = d(250 - 2d) \) is 10 ft.
• A reasonable scale for the vertical axis of the function \( A(d) = d(250 - 2d) \) is 2000 sq ft.

Answer 2

Question
17. Complete the statements.(2 points)
The product of 6⋅3–√
represents the product of
a rational
number and an irrational number. The result of the product of 6⋅3–√
is
an irrational
number

Answer 3

To complete the statements regarding the expression \( 6 \cdot \sqrt{3} \):

1. The product of \( 6 \cdot \sqrt{3} \) represents the product of:

   • a rational number (6) and an irrational number (\( \sqrt{3} \)).

2. The result of the product of \( 6 \cdot \sqrt{3} \) is:

   • an irrational number.

Putting it all together, the completed statements are:

• The product of \( 6 \cdot \sqrt{3} \) represents the product of a rational number and an irrational number.
• The result of the product of \( 6 \cdot \sqrt{3} \) is an irrational number.