The foot of a ladder is on level ground 1.5m from a wall. The ladder leans agents the wall. The angle formed by the ladder and the ground is 70 degrees. Calculate how high up the wall the ladder reaches.
To find out how high up the wall the ladder reaches, we can use trigonometry.
First, let's label the information we have:
- The distance from the foot of the ladder to the wall is given as 1.5 meters.
- The angle formed between the ladder and the ground is 70 degrees.
We need to find the height up the wall that the ladder reaches. This can be calculated using the sine function:
sin(angle) = opposite / hypotenuse
In this case, the opposite side (height) is what we're trying to find, and the hypotenuse is the length of the ladder.
We can rearrange the formula to solve for the height:
height = sin(angle) * hypotenuse
Since we have the angle and the distance from the foot of the ladder to the wall, we can use these values to find the height.
height = sin(70 degrees) * 1.5 meters
Using a calculator, we find that sin(70 degrees) is approximately 0.9397.
height ≈ 0.9397 * 1.5
height ≈ 1.409 meters
So, the ladder reaches a height of approximately 1.409 meters up the wall.
To calculate how high up the wall the ladder reaches, we can use trigonometry. In this case, we have a right triangle formed by the ladder, the ground, and the wall.
Let's call the height up the wall that the ladder reaches "h" (in meters). We already know that the distance from the foot of the ladder to the wall is 1.5m, and the angle between the ground and the ladder is 70 degrees.
To find the height, we can use the sine function. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse (in this case, the hypotenuse is the ladder).
Therefore, we can write the equation:
sin(70 degrees) = h / length of the ladder
To solve for h, we need to find the length of the ladder. The length of the ladder can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the other two sides are the distance from the foot of the ladder to the wall (1.5m) and the height h.
So, the equation for the length of the ladder (L) is:
L^2 = 1.5^2 + h^2
To find h, we can rearrange the equation to solve for h:
h^2 = L^2 - 1.5^2
Now that we have an equation for h^2, we can substitute it into the original equation involving sine:
sin(70 degrees) = h / L
sin(70 degrees) = √(L^2 - 1.5^2) / L
We can rearrange this equation to solve for h:
h = L * sin(70 degrees)
To calculate h, we need to find the value of sin(70 degrees). Using a calculator, we get sin(70 degrees) ≈ 0.9397.
Now, let's calculate h:
h = L * 0.9397
To find L, we can substitute the value of h into the Pythagorean equation we derived earlier:
h^2 = L^2 - 1.5^2
Now we can solve for L:
L = √(h^2 + 1.5^2)
Substituting the value of h into this equation will give us the length of the ladder.
153
tan 70 = height/1.5
height = 1.5 tan 70°
= ...