Nicholas places his 30-foot ladder against a house he is painting. If the foot of the ladder is 8 feet from the base of the house, how high above the ground is the top of the ladder touching the house, to the nearest tenth of a foot?(1 point)
Responses
22.0 ft.
22.0 ft.
31.0 ft.
31.0 ft.
28.9 ft.
28.9 ft.
27.8 ft.
To determine the height above the ground that the top of the ladder is touching the house, we can use the Pythagorean theorem.
The ladder is the hypotenuse of a right triangle, and the base of the ladder is one of the legs. The height above the ground that the top of the ladder is touching the house is the other leg.
Using the Pythagorean theorem, we have:
height^2 + base^2 = ladder^2.
Let "h" be the height above the ground that the top of the ladder is touching the house. Then, we have:
h^2 + 8^2 = 30^2.
Simplifying the equation:
h^2 + 64 = 900.
Subtracting 64 from both sides:
h^2 = 836.
Taking the square root of both sides:
h ≈ √836 ≈ 28.9 feet.
Therefore, the height above the ground that the top of the ladder is touching the house is approximately 28.9 feet, to the nearest tenth.
Use the figure to answer the question.
Lines P and Q are parallel. Which of the following pairs of angles have the same measure?
(1 point)
Responses
∠5
and ∠6
angle 5 and angle 6
∠6
and ∠7
angle 6 and angle 7
∠2
and ∠3
angle 2 and angle 3
∠1
and ∠8
In the given figure where lines P and Q are parallel, we can see that they are intersected by a transversal which forms multiple pairs of angles.
By the corresponding angles postulate, angles in corresponding positions relative to the parallel lines are congruent.
From the figure, we can see that ∠5 and ∠6 are in corresponding positions, meaning they are formed by the transversal intersecting line P and Q.
Therefore, ∠5 and ∠6 have the same measure.
The frozen yogurt cone shown is made from a cone and a hemisphere. The radius is 6 cm. The total height of the cone is 24 cm. The cone is completely packed with frozen yogurt.
What is the exact volume of the frozen yogurt in the cone and hemisphere? Leave your answer in terms of pi.
(1 point)
Responses
288π cm3
288π cm3
360π cm3
360π cm3
144π cm3
144π cm3
216π cm3
To find the volume of the frozen yogurt in the cone and hemisphere, we need to find the volume of each shape separately and then add them together.
The volume of a cone is given by the formula: V = (1/3)πr^2h, where r is the radius and h is the height.
We are given that the radius of the cone is 6 cm and the total height is 24 cm. Plugging these values into the formula, we get:
V_cone = (1/3)π(6^2)(24) = 288π cm^3.
The volume of a hemisphere is given by the formula: V = (2/3)πr^3.
Since the radius is also 6 cm, we can calculate the volume of the hemisphere:
V_hemisphere = (2/3)π(6^3) = 288π cm^3.
To find the total volume of the frozen yogurt, we add the volume of the cone and the volume of the hemisphere:
Total volume = V_cone + V_hemisphere = 288π + 288π = 576π cm^3.
Therefore, the exact volume of the frozen yogurt in the cone and hemisphere is 576π cm^3.