Two ladders are the same distance from the base of a wall. The longer ladder is 15m long and makes an angle of 58 with the ground. if the shorter ladder is 12.6m long, what angle does it make with the ground?
d = 15*cos58 = 7.95 m. = Distance from
base of the wall.
Cos A = d/L = 7.95/12.6 = 0.63086
A = 50.9o
To determine the angle the shorter ladder makes with the ground, we can use the concept of trigonometry. In this case, we can use the tangent function.
Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.
Given that the longer ladder is 15m long and makes an angle of 58 degrees with the ground, we can label the opposite side as "opposite" and the adjacent side as "adjacent".
Using the tangent function:
tan(angle) = opposite / adjacent
tan(58) = opposite / 15
To find the value of the opposite side:
opposite = tan(58) * 15
Now, we can find the angle the shorter ladder makes with the ground using the shorter ladder's length and the calculated value of the opposite side.
tan(angle) = opposite / adjacent
tan(angle) = opposite / 12.6
Rearranging the formula to solve for the angle:
angle = atan(opposite / adjacent)
angle = atan((tan(58) * 15) / 12.6)
Using a calculator, we can evaluate the expression:
angle ≈ atan(0.8433) ≈ 40.9 degrees
Therefore, the shorter ladder makes an angle of approximately 40.9 degrees with the ground.
To find the angle that the shorter ladder makes with the ground, we can use the concept of similar triangles. Let's assume that the shorter ladder makes an angle θ with the ground.
First, let's consider the longer ladder. We know its length is 15m and it forms an angle of 58 degrees with the ground. We can label the base of the ladder as 'x' (the distance from the base of the wall), and the height as 'y' (the height it reaches on the wall).
Using trigonometry, we can write the following equation:
sin(58 degrees) = y / 15m.
Now, let's consider the shorter ladder. We know its length is 12.6m and it forms an angle of θ with the ground. Again, we can label the base of the ladder as 'x' (the distance from the base of the wall), and the height as 'y' (the height it reaches on the wall).
Using the concept of similar triangles, the ratios of the corresponding sides of similar triangles are equal. Therefore, we can set up the following equation:
y / x = (y / 15m) * (12.6m / 15m).
Simplifying this equation, we get:
x = (15m * y) / (12.6m).
We can now substitute the value of x from this equation into the first equation:
sin(58 degrees) = y / ((15m * y) / (12.6m)).
To find the value of y, we can cross-multiply and solve the equation:
sin(58 degrees) * ((15m * y) / (12.6m)) = y.
Simplifying this equation, we get:
y = (sin(58 degrees) * 15m * y) / (12.6m).
To solve this equation for y, we can divide both sides by (sin(58 degrees) * 15m / 12.6m):
y / ((sin(58 degrees) * 15m) / 12.6m) = 1.
Now, we can calculate the value of y:
y = ((sin(58 degrees) * 15m) / 12.6m).
Finally, we can use this value of y to find the angle θ that the shorter ladder makes with the ground:
tan(θ) = y / x.
Substituting the value of y ((sin(58 degrees) * 15m) / 12.6m) and the length of the shorter ladder (12.6m), we get:
tan(θ) = ((sin(58 degrees) * 15m) / 12.6m) / 12.6m.
Solving this equation for θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(((sin(58 degrees) * 15m) / 12.6m) / 12.6m).
Using a calculator, we can evaluate this expression to get the angle θ in degrees.