Question

The sum of three consecutive odd integers is 39. List the integers, from least to greatest.

Answer 1

The area of a trapezoid is found with the formula A=12(b1+b2)h.
A trapezoidal banner has an area of 120 m² and a height of 10 m. Base 1 (b1) is 16 m shorter than base 2 (b2). What are the lengths of the two bases? Let b represent the length of base 1.

Answer 2

A rectangle has an area of 102 cm2. The length of the rectangle is 17 cm.

What is the perimeter of the rectangle?

Answer 3

Let the three consecutive odd integers be represented as \( x \), \( x + 2 \), and \( x + 4 \).

According to the problem, the sum of these integers is equal to 39:

\[
x + (x + 2) + (x + 4) = 39
\]

Combining the terms, we have:

\[
3x + 6 = 39
\]

Next, we subtract 6 from both sides:

\[
3x = 33
\]

Now, we divide both sides by 3:

\[
x = 11
\]

Now we can find the three consecutive odd integers:

- The first integer is \( x = 11 \)
- The second integer is \( x + 2 = 13 \)
- The third integer is \( x + 4 = 15 \)

Thus, the three consecutive odd integers are \( 11, 13, \) and \( 15 \).

From least to greatest, the integers are:

\[
\boxed{11, 13, 15}
\]

Answer 4

To find the lengths of the two bases of the trapezoid, we can use the given formula for the area:

\[
A = \frac{1}{2} (b_1 + b_2) h
\]

Given that the area \( A = 120 \) m² and the height \( h = 10 \) m, we substitute these values into the area formula:

\[
120 = \frac{1}{2} (b_1 + b_2) \times 10
\]

We can simplify this equation:

\[
120 = 5 (b_1 + b_2)
\]

Now, divide both sides by 5:

\[
b_1 + b_2 = 24
\]

Next, we are told that base 1 (\( b_1 \)) is 16 m shorter than base 2 (\( b_2 \)). Therefore, we can express \( b_1 \) in terms of \( b_2 \):

\[
b_1 = b_2 - 16
\]

Now we can substitute this expression for \( b_1 \) into the equation \( b_1 + b_2 = 24 \):

\[
(b_2 - 16) + b_2 = 24
\]

Combining like terms gives us:

\[
2b_2 - 16 = 24
\]

Next, we add 16 to both sides:

\[
2b_2 = 40
\]

Now, divide both sides by 2:

\[
b_2 = 20
\]

Now that we have \( b_2 \), we can find \( b_1 \):

\[
b_1 = b_2 - 16 = 20 - 16 = 4
\]

Thus, the lengths of the two bases are:

- Base 1, \( b_1 = 4 \) m
- Base 2, \( b_2 = 20 \) m

In conclusion, the lengths of the bases are:

\[
\boxed{4 \text{ m} \text{ and } 20 \text{ m}}
\]

Answer 5

To find the perimeter of the rectangle, we first need to determine its width using the area formula for a rectangle, which is:

\[
\text{Area} = \text{length} \times \text{width}
\]

Given that the area is \(102 \, \text{cm}^2\) and the length is \(17 \, \text{cm}\), we can set up the equation:

\[
102 = 17 \times \text{width}
\]

To find the width, we divide both sides by \(17\):

\[
\text{width} = \frac{102}{17} = 6 \, \text{cm}
\]

Now that we have both the length and the width of the rectangle, we can calculate the perimeter using the formula for the perimeter of a rectangle:

\[
\text{Perimeter} = 2 \times (\text{length} + \text{width})
\]

Substituting the values we found:

\[
\text{Perimeter} = 2 \times (17 + 6) = 2 \times 23 = 46 \, \text{cm}
\]

Therefore, the perimeter of the rectangle is:

\[
\boxed{46 \, \text{cm}}
\]