The Dig It Project is
designing two gardens that have the same perimeter.
One garden is a trapezoid whose nonparallel sides are
equal. The other is a quadrilateral. Two possible designs
are shown at the right.
a. Based on these designs, is there more than one
value for x? Explain how you know this.
b. Why does your answer to part a make sense in this context?
c. Suppose the Dig It Project wants the perimeter of each garden to be
60 meters. What is the value of x in this case? How did you find this?
No designs shown. Cannot copy and paste here.
No
a. Based on the designs shown, there is only one value for x. In both designs, x represents the length of one side of the trapezoid, and it is clear that x is the same length in both designs.
b. This makes sense in the context because the question states that the two gardens have the same perimeter. To achieve this, the length of x, which is the nonparallel side of the trapezoid, must be equal in both designs.
c. If the perimeter of each garden is 60 meters, we can set up an equation based on the given designs. In both designs, we have one side equal to x and two sides equal to 20 meters. Additionally, in the first design, we have two sides equal to 10 meters each, and in the second design, we have one side equal to 10 meters and one side equal to 5 meters.
Using the equation for the perimeter (P = sum of all sides), we can solve for x:
1st design: P = x + x + 20 + 20 + 10 + 10 = 60 meters
2x + 60 = 60
2x = 60 - 60
2x = 0
x = 0/2
x = 0 meters
2nd design: P = x + 20 + 20 + 10 + 10 + 5 = 60 meters
x + 65 = 60
x = 60 - 65
x = -5 meters
Therefore, the value of x in this case is 0 meters. We found this by setting up our equations and solving for x. The negative value of x for the second design is not feasible in this context as it represents an invalid measurement for a side length.
a. In order to determine if there is more than one value for x, we need to analyze the designs of the two gardens and identify any constraints on the variable x.
Looking at the first garden design, it is a trapezoid with nonparallel sides being equal. Let's label the lengths of the sides: the parallel base is 12 meters, one of the nonparallel sides is x meters, and the other nonparallel side is also x meters. The remaining side, called the height, is not specified.
In the second garden design, it is a quadrilateral with sides of 15 meters, 12 meters, and x meters. The fourth side is not given a specific length.
To determine if there is more than one value for x, we need to examine if there are any constraints on x in each design.
In the first garden design, the nonparallel sides are equal in length. Therefore, the value of x must be the same for both nonparallel sides. This means there is only a single value for x in this design.
In the second garden design, there are no constraints explicitly mentioned that would require x to have a specific value. Hence, there could be multiple values for x in this design.
b. The answer to part a makes sense in this context because the question asks if there is more than one value for x based on the given designs. It is possible to have designs that restrict the variable to a single value and others that allow for multiple values. The uniqueness of x in the first design could be due to the symmetry required for the equal nonparallel sides of a trapezoid. On the other hand, the second design does not have any explicit constraints, so there is more flexibility in determining values for x.
c. Suppose the Dig It Project wants the perimeter of each garden to be 60 meters. In this case, we can use the given information to find the value of x in each design.
For the first design, we know that one parallel base has a length of 12 meters and both nonparallel sides have a length of x meters. To find the value of x, we can set up the equation:
12 + x + x + H = 60,
where H represents the height. Simplifying, we get:
2x + H = 60 - 12,
2x + H = 48.
Since we don't have enough information to determine the height H, we cannot find a specific value for x in this case.
For the second design, we have side lengths of 15 meters, 12 meters, and x meters. Using the perimeter equation:
15 + 12 + x + Y = 60,
where Y represents the fourth side, we can simplify to:
27 + x + Y = 60,
x + Y = 60 - 27,
x + Y = 33.
Again, without a specified value for Y, we cannot determine a specific value for x in this case either.
Therefore, in both designs, there is no unique value for x when the perimeter is 60 meters because the additional side lengths or heights are not given, preventing us from calculating a specific value for x.