Question

A straw is placed inside a rectangular box that is 10 inches by 8 inches by 9 inches, as shown. If the straw fits exactly into the box diagonally from the bottom left corner to the top right back corner, how long is the straw? Leave your answer in simplest radical form.

Answer 1

[AI-generated answer — Bot GPT 3.5]

The diagonal of a rectangular box can be found using the formula:

diagonal = √(length^2 + width^2 + height^2)

Plugging in the values:
diagonal = √(10^2 + 8^2 + 9^2)
diagonal = √(100 + 64 + 81)
diagonal = √245
diagonal = 7√5

Therefore, the length of the straw is 7√5 inches.

Answer 2

One of the legs of a right triangle measures 2 cm and the other leg measures 7 cm. Find the measure of the hypotenuse. If necessary, round to the nearest tenth.

Answer 3

[AI-generated answer — Bot GPT 3.5]

Using the Pythagorean theorem, we can find the length of the hypotenuse:

hypotenuse^2 = leg1^2 + leg2^2
hypotenuse^2 = 2^2 + 7^2
hypotenuse^2 = 4 + 49
hypotenuse^2 = 53

Taking the square root of both sides to find hypotenuse:
hypotenuse = √53
hypotenuse ≈ 7.3 cm

Therefore, the length of the hypotenuse is approximately 7.3 cm.

Answer 4

One of the legs of a right triangle measures 7 cm and the other leg measures 4 cm. Find the measure of the hypotenuse. If necessary, round to the nearest tenth.

Answer 5

[AI-generated answer — Bot GPT 3.5]

Using the Pythagorean theorem, we can find the length of the hypotenuse:

hypotenuse^2 = leg1^2 + leg2^2
hypotenuse^2 = 7^2 + 4^2
hypotenuse^2 = 49 + 16
hypotenuse^2 = 65

Taking the square root of both sides to find the hypotenuse:
hypotenuse = √65
hypotenuse ≈ 8.1 cm

Therefore, the length of the hypotenuse is approximately 8.1 cm.

Answer 6

the first side is 25 and the second side is 7.Find the length of the third side. of the triangle

Answer 7

[AI-generated answer — Bot GPT 3.5]

If the first side of the triangle is 25 and the second side is 7, the third side can be found using the Pythagorean theorem.

Let's denote the third side as x.

Using the Pythagorean theorem:

25^2 = 7^2 + x^2
625 = 49 + x^2
625 - 49 = x^2
576 = x^2

Taking the square root of both sides to solve for x:

x = √576
x = 24

Therefore, the length of the third side is 24.

Answer 8

An altitude is drawn from the vertex of an isosceles triangle, forming a right angle and two congruent triangles. As a result, the altitude cuts the base into two equal segments. The length of the altitude is 11 inches, and the length of the base is 6 inches. Find the triangle’s perimeter. Round to the nearest tenth of an inch.

Answer 9

[AI-generated answer — Bot GPT 3.5]

In an isosceles triangle where an altitude is drawn from the vertex to the midpoint of the base, it creates two right triangles which are also congruent.

Let's denote the height (altitude) as h = 11 inches and the base as b = 6 inches.

The altitude divides the base into two equal segments of 3 inches each.

We can find the length of the legs of the right triangle formed using the Pythagorean theorem:
Legs = √(altitude^2 + (1/2 * base)^2)

The length of the legs in the right triangle is:
Legs = √(11^2 + 3^2)
Legs = √(121 + 9)
Legs = √130

The perimeter of the triangle is:
Perimeter = 2 * Leg + Base
Perimeter = 2 * √130 + 6
Perimeter ≈ 2 * 11.4 + 6
Perimeter ≈ 22.8 + 6
Perimeter ≈ 28.8 inches

Therefore, the triangle's perimeter is approximately 28.8 inches.

Answer 10

A 30 foot ladder is set against the side of a house so that it reaches up 24 feet. If Alexander grabs the ladder at its base and pulls it 10 feet farther from the house, how far up the side of the house will the ladder reach now? (The answer is not 14 ft.) Round to the nearest tenth of a foot.

Answer 11

[AI-generated answer — Bot GPT 3.5]

When the ladder is set against the house originally, it forms a right triangle with the house wall and the ground.

The ladder (hypotenuse) is 30 feet and the height it reaches is 24 feet. We can use the Pythagorean theorem to find the distance of the ladder from the base to the top of the house wall.

Let x be the distance that the ladder moves horizontally away from the house.

Using the Pythagorean theorem:

(24 + 10)^2 = x^2 + 30^2
34^2 = x^2 + 900
1156 = x^2 + 900
x^2 = 1156 - 900
x^2 = 256
x = √256
x = 16 feet

So, when Alexander pulls the ladder 10 feet farther from the house, it will reach 16 feet up the side of the house.