The perimeter of a scalene triangle is 14.5 cm. The longest side is twice that of the shortest side. Which equation can be used to find the side lengths if the longest side measures 6.2 cm?

Question

A scalene triangle where all sides are of different lengths. The base of the triangle is labeled 2a, is twice that of the shortest side labeled a. The other side is labeled b.

Choose the correct answer.
6.2+b=14.5
9.3+b=14.5
12.4+b=14.5
18.6+b=14.5

Answer 1

Let's denote the shortest side as \( a \). According to the information provided:

• The longest side \( l \) is \( 2a \).
• The longest side is given as \( 6.2 \) cm.

Since \( 2a = 6.2 \), we can solve for \( a \):

\[ a = \frac{6.2}{2} = 3.1 \text{ cm} \]

Now, we have:

• Shortest side \( a = 3.1 \) cm
• Longest side \( l = 6.2 \) cm

Let the other side be \( b \). The perimeter of the triangle is given to be \( 14.5 \) cm. The equation for the perimeter can be set up as:

\[ a + b + l = 14.5 \]

Substituting the values we have for \( a \) and \( l \):

\[ 3.1 + b + 6.2 = 14.5 \]

Now, let’s simplify this equation:

\[ 9.3 + b = 14.5 \]

From here, we can see that the correct equation to find the side lengths is:

\[ 9.3 + b = 14.5 \]

The correct answer is: 9.3 + b = 14.5.

Answer 2

A 2-column table with 5 rows. The first column, number of snapdragons, x, has the entries 11, 12, 13, 14. The second column, number of daisies, y, has the entries 34, 33, 32, 31.
Hans is planting a garden with snapdragons and daisies. The table shows some possible combinations of the two plants. If Hans plants 29 daisies, how many snapdragons will he plant?

The equation_________
models the scenario.

Hans will plant_________
snapdragons.

Answer 3

To model the relationship between the number of snapdragons \( x \) and the number of daisies \( y \) based on the table, we can observe the data:

\[ \begin{array}{|c|c|} \hline \text{Number of Snapdragons (x)} & \text{Number of Daisies (y)} \ \hline 11 & 34 \ 12 & 33 \ 13 & 32 \ 14 & 31 \ \hline \end{array} \]

From the table, we can see that as the number of snapdragons increases by 1, the number of daisies decreases by 1. This suggests a linear relationship.

We can express this relationship in the form of an equation. Notice that the pattern suggests:

\[ y = 45 - x \]

This equation models the scenario, where \( 45 \) is a constant that adjusts the relationship based on the totals in the table (it can be derived by adding the maximum entries \( x + y \) when \( x = 14, y = 31 \)).

Next, we need to determine how many snapdragons Hans will plant if he decides to plant 29 daisies. We set \( y = 29 \) in the equation:

\[ 29 = 45 - x \]

To find \( x \), we can rearrange the equation:

\[ x = 45 - 29 \] \[ x = 16 \]

Therefore, Hans will plant:

The equation \( y = 45 - x \) models the scenario.

Hans will plant 16 snapdragons.

Answer 4

Hans is planting a garden with snapdragons and daisies. The table shows some possible combinations of the two plants. If Hans plants 29 daisies, how many snapdragons will he plant?

The equation
x = 45 – y
models the scenario.

Hans will plant
16
snapdragons.

What is the value of x in the equation 3 x minus 4 y equals 65, when y equals 4?

x equals 13 and one-fourth
x equals 21 and two-thirds

x=23
x=27

Answer 5

To find the value of \( x \) in the equation \( 3x - 4y = 65 \) when \( y = 4 \), we will substitute \( y \) with \( 4 \) in the equation:

\[ 3x - 4(4) = 65 \]

Now, simplify it:

\[ 3x - 16 = 65 \]

Next, we add \( 16 \) to both sides:

\[ 3x = 65 + 16 \]

\[ 3x = 81 \]

Now, divide both sides by \( 3 \):

\[ x = \frac{81}{3} = 27 \]

Therefore, the value of \( x \) is:

\( x = 27 \)

So the correct answer is:

x = 27.

Answer 6

An isosceles triangle has two sides of equal length, a, and a base, b. The perimeter of the triangle is 15.7 inches, so the equation to solve is 2a + b = 15.7.

If we recall that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, which lengths make sense for possible values of b? Select two options.

–2 in.
0 in.
0.5 in.
2 in.
7.9 in.

Answer 7

To determine the possible values of \( b \) (the base of the isosceles triangle) based on the perimeter and the triangle inequality theorem, we start with the given equation:

\[ 2a + b = 15.7 \]

From this equation, we can express \( b \) in terms of \( a \):

\[ b = 15.7 - 2a \]

Next, we need to apply the triangle inequalities for an isosceles triangle, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

In our scenario, the inequalities that must be satisfied are:

1. \( a + a > b \) (which simplifies to \( 2a > b \))
2. \( a + b > a \) (which simplifies to \( b > 0 \))
3. \( a + b > a \) (which simplifies again to \( b > 0 \))

Since the second and third inequalities are identical, we can focus on them along with the first inequality.

From the first inequality \( 2a > b \), we substitute \( b \) from our expression:

\[ 2a > 15.7 - 2a \]

Adding \( 2a \) to both sides gives:

\[ 4a > 15.7 \]

Dividing by \( 4 \):

\[ a > \frac{15.7}{4} = 3.925 \]

Now we can substitute this value of \( a \) back into the equation for \( b \):

\[ b = 15.7 - 2a \]

If \( a \) is slightly more than \( 3.925 \), we can find the maximum value for \( b \) by using the smallest possible integer value for \( a \):

Assume \( a = 3.93 \):

\[ b = 15.7 - 2(3.93) = 15.7 - 7.86 = 7.84 \]

Thus, we already know \( b \) must be less than \( 7.84 \).

Given that \( b > 0 \), the possible values for \( b \) from the options provided are:

• 0.5 in. (since \( b > 0 \))
• 2 in. (since \( b < 7.84 \))

The other options either do not satisfy \( b > 0 \) or exceed \( 7.84 \) (like \( 7.9 \) in.).

Thus, the two lengths that make sense for possible values of \( b \) are:

• 0.5 in.
• 2 in.

Answer 8

Shana wants to use all 62 feet of the fencing she has to make a rectangular run for her dog. She decides to make the length of the run 20 feet. She writes and solves the equation 2 l plus 2 w equals 62 to find the width of the run.

Which statements are true of the solution? Check all that apply.

The value of w is 10 feet.
The value of w can be zero.
The value of w cannot be a negative number.
Substitution is used to replace the variable l with a value of 20.
The subtraction property of equality is used to isolate the term with the variable w.

Answer 9

To analyze the situation and validate the statements, let's start with the equation Shana wrote:

\[ 2l + 2w = 62 \]

Given that the length \( l \) is 20 feet, we can substitute \( l = 20 \) into the equation:

\[ 2(20) + 2w = 62 \]

This simplifies to:

\[ 40 + 2w = 62 \]

Next, we isolate \( 2w \) by subtracting 40 from both sides:

\[ 2w = 62 - 40 \] \[ 2w = 22 \]

Now, we divide by 2 to find \( w \):

\[ w = \frac{22}{2} = 11 \]

Now that we have found the value of \( w \) (which is 11 feet), we can evaluate the truth of each statement:

1. The value of w is 10 feet.
   False. We calculated \( w \) to be 11 feet.

2. The value of w can be zero.
   False. While in theory, a width of zero is a solution to a linear equation, it does not make sense physically for a fencing example – the width of a rectangular run must be greater than zero to have an area.

3. The value of w cannot be a negative number.
   True. Width cannot be negative in the context of a rectangular run.

4. Substitution is used to replace the variable l with a value of 20.
   True. This is correct, as we substituted \( l \) with 20 feet in the original equation.

5. The subtraction property of equality is used to isolate the term with the variable w.
   True. We used subtraction to isolate the term \( 2w \) after substituting \( l \).

In summary, the true statements are:

• The value of w cannot be a negative number.
• Substitution is used to replace the variable l with a value of 20.
• The subtraction property of equality is used to isolate the term with the variable w.

Answer 10

Jillian’s school is selling tickets for a play. The tickets cost $10.50 for adults and $3.75 for students. The ticket sales for opening night totaled $2071.50. The equation 10.50 a plus 3.75 b equals 2071.50, where a is the number of adult tickets sold and b is the number of student tickets sold, can be used to find the number of adult and student tickets. If 82 students attended, how may adult tickets were sold?

adult tickets

Answer 11

To find the number of adult tickets sold, we start with the equation representing the total ticket sales, given by:

\[ 10.50a + 3.75b = 2071.50 \]

where:

• \( a \) is the number of adult tickets sold,
• \( b \) is the number of student tickets sold.

We're given that \( b = 82 \) (the number of student tickets sold). We can substitute this value into the equation:

\[ 10.50a + 3.75(82) = 2071.50 \]

Next, calculate \( 3.75 \times 82 \):

\[ 3.75 \times 82 = 307.50 \]

Now, substitute this value back into the equation:

\[ 10.50a + 307.50 = 2071.50 \]

Next, isolate the term with \( a \) by subtracting \( 307.50 \) from both sides:

\[ 10.50a = 2071.50 - 307.50 \]

This simplifies to:

\[ 10.50a = 1764 \]

Now, divide both sides by \( 10.50 \) to solve for \( a \):

\[ a = \frac{1764}{10.50} \]

Calculating this gives:

\[ a = 168 \]

Thus, the number of adult tickets sold is:

168 adult tickets.