The height of an equilateral triangle equals 3√2

Question

times the side length. Using 74
for the square root of 3, what is the approximate height in meters of an equilateral triangle with a side length of 8 meters?(1 point)
$$
Skip to navigation

Answer 1

To find the height of an equilateral triangle in terms of its side length, we can use the formula for the height \(h\) of an equilateral triangle, which is given by:

\[ h = \frac{\sqrt{3}}{2} s \]

where \(s\) is the side length of the triangle.

According to the problem, the height is also said to be equal to \(3\sqrt{2}\) times the side length, which gives us another form:

\[ h = 3\sqrt{2} \cdot s \]

Let’s use the first formula since it follows the standard calculation for an equilateral triangle. You've given that we should use \(74\) for \(\sqrt{3}\). So we can proceed with the calculation.

Given that the side length \(s = 8\) meters, we first calculate the height using the provided approximation for \(\sqrt{3}\):

\[ h = \frac{\sqrt{3}}{2} s = \frac{74}{2} \cdot 8 \]

Calculating this step by step:

1. First, calculate \(\frac{74}{2}\):

\[ \frac{74}{2} = 37 \]

2. Now multiply by the side length \(s = 8\):

\[ h = 37 \cdot 8 = 296 \]

Thus, the approximate height of the equilateral triangle with a side length of 8 meters is:

\[ \boxed{296 \text{ meters}} \]

Answer 2

The diagonal of a square equals 2–√
times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)

Answer 3

To find the length of the diagonal of a square in terms of its side length, we can use the formula:

\[ d = \sqrt{2} \cdot s \]

where \(d\) is the diagonal length and \(s\) is the side length.

Given that we should use \(1.4\) for \(\sqrt{2}\), and the side length \(s = 700\) feet, we can substitute these values into the formula.

So, using the formula:

\[ d = \sqrt{2} \cdot s = 1.4 \cdot 700 \]

Now, let's calculate the diagonal:

\[ d = 1.4 \cdot 700 = 980 \]

Thus, the approximate length from one corner to an opposite corner of the square park is:

\[ \boxed{980 \text{ feet}} \]

Answer 4

Using a number line, what whole number and rational number to the nearest half is 40−−√
between?(1 point)
40−−√
is between
and
.

Answer 5

To find the whole number and the rational number to the nearest half that \( \sqrt{40} \) is between, we first need to calculate \( \sqrt{40} \).

We know that \( \sqrt{40} \) can be simplified as follows:

\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]

Next, we need to find the approximate value of \( \sqrt{10} \). The approximate value of \( \sqrt{10} \) is about \( 3.16 \). Therefore:

\[ \sqrt{40} \approx 2 \times 3.16 \approx 6.32 \]

Now, we can deduce the whole number and the rational number to the nearest half surrounding \( \sqrt{40} \):

• The whole number that \( \sqrt{40} \) is between is 6 and 7. Since \( 6.32 \) is between these two whole numbers.

• The nearest half around \( 6.32 \) would be 6.5.

Thus, \( \sqrt{40} \) is between \( 6 \) and \( 6.5 \).

So the answer is:

\(\sqrt{40}\) is between 6 and 6.5.

Answer 6

Using a number line, what whole number and rational number to the nearest half is 58−−√
between?(1 point)
Responses

between 8.5 and 9
between 8.5 and 9

between 7 and 7.5
between 7 and 7.5

between 7.5 and 8
between 7.5 and 8

between 8 and 8.5

Answer 7

To determine the whole number and rational number to the nearest half that \( \sqrt{58} \) is between, we first need to calculate \( \sqrt{58} \).

We know that:

• \( 7^2 = 49 \)
• \( 8^2 = 64 \)

Thus, we can see that:

\[ 7 < \sqrt{58} < 8 \]

Now, to get a more precise estimation, we can calculate:

\[ \sqrt{58} \approx 7.62 \]

Now let's find where \( \sqrt{58} \) falls regarding whole numbers and half numbers:

Since \( 7.62 \) is between:

• 7 and 8, but closer to 8.

For the nearest half, it would be between 7.5 and 8.

Thus, \( \sqrt{58} \) is approximately:

• between 7.5 and 8.

So the correct response would be:

between 7.5 and 8.

Answer 8

Using rational approximations, what is an inequality statement comparing 81−−√
and 8–√8
? Enter <, >, or =.(1 point)
The inequality is: 81−−√

8–√8
.

Answer 9

To compare \( \sqrt{81} \) and \( 8 - \sqrt{8} \), let's calculate both expressions:

1. Calculating \( \sqrt{81} \): \[ \sqrt{81} = 9 \]

2. Calculating \( 8 - \sqrt{8} \): First, let's approximate \( \sqrt{8} \). \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \]

   Now, substitute this approximation into \( 8 - \sqrt{8} \): \[ 8 - \sqrt{8} \approx 8 - 2.828 \approx 5.172 \]

Now we can compare: \[ 9 \quad \text{and} \quad 5.172 \]

Since \( 9 > 5.172 \), we can conclude that: \[ \sqrt{81} > 8 - \sqrt{8} \]

Thus, the inequality statement is: \[ \sqrt{81} > 8 - \sqrt{8} \]

The answer is: \[

\]

Answer 10

Write 6,739 in expanded form.(1 point)
Responses

(6×1,000)+(7×100)+(3×10)+(0×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 0 times 1 right parenthesis

(9×1,000)+(3×100)+(7×10)+(6×1)
left parenthesis 9 times 1,000 right parenthesis plus left parenthesis 3 times 100 right parenthesis plus left parenthesis 7 times 10 right parenthesis plus left parenthesis 6 times 1 right parenthesis

(7×100)+(3×10)+(9×1)
left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis

(6×1,000)+(7×100)+(3×10)+(9×1)

Answer 11

To write the number 6,739 in expanded form, we break it down by each digit based on its place value:

• \(6\) is in the thousands place, so it represents \(6 \times 1,000\).
• \(7\) is in the hundreds place, so it represents \(7 \times 100\).
• \(3\) is in the tens place, so it represents \(3 \times 10\).
• \(9\) is in the ones place, so it represents \(9 \times 1\).

Putting it all together, the expanded form of 6,739 is:

\[ (6 \times 1,000) + (7 \times 100) + (3 \times 10) + (9 \times 1) \]

Therefore, the correct answer is:

\[ (6\times1,000) + (7\times100) + (3\times10) + (9\times1) \]