A 40-N box is suspended in the ceilingusing two ropes . Each ropes makes an angle of 45° with the vertical . How much is the tension in each rope?

two conditions. break the ropes into verical and horizontal componednts.

Horizontal:
Fl*sinthetaL-FR*sinThetaR=0 (left and right)
Vertical
Fl*cosThetal+Fr*cosTehtaR=40 (sum of forces equal weight).

well, your angles makes it easy, all the trig functions are .707
so immediately, from the horizontal equation, we have Fr=Fl in magnitude

and the vertical
fl=fr=40/.707

T1*sin(90+45) + T2*sin45 = 40.

0.707T1 + 0.707T2 = 40.
T1 = T2. Replace T2 with T1:
0.707T1 + 0.707T1 = 40.
T1 = 28.3 N. = T2.

To find the tension in each rope, we need to decompose the weight of the box into its vertical and horizontal components.

Given that the box weighs 40 N and the ropes make an angle of 45° with the vertical, we can determine the vertical and horizontal components using trigonometry. The vertical component represents the force pulling directly upwards, while the horizontal component represents the force pulling sideways.

The vertical component of the weight can be found by multiplying the weight by the cosine of the angle, since cosine gives us the ratio of the adjacent side (vertical component) to the hypotenuse (weight). So, the vertical component is:

Vertical component = Weight * cos(angle)
= 40 N * cos(45°)

Similarly, the horizontal component can be found by multiplying the weight by the sine of the angle, since sine gives us the ratio of the opposite side (horizontal component) to the hypotenuse (weight). So, the horizontal component is:

Horizontal component = Weight * sin(angle)
= 40 N * sin(45°)

Since the box is in equilibrium, the tension in each rope must balance out the vertical and horizontal components. Thus, the tension in each rope is equal to the magnitudes of these components, which means that both ropes have the same tension.

Therefore, the tension in each rope is:

Tension = Vertical component
= 40 N * cos(45°)

Calculating this value, we get:

Tension = 40 N * 0.7071 ≈ 28.28 N

So, the tension in each rope is approximately 28.28 N.