Two forces with magnitudes of 200 and 100 pounds act on an object at angles of 60° and 170°, respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer. Please show work step by step. Step by step ONLY. If step by step is not shown it shall be reported.

  190.79 pounds at 88.11°   or   191.01 pounds at 89.40°

Step-by-step explanation:

The effect of rounding will be different, depending on the method you choose for solving the problem. In any event, rounding intermediate results is an inappropriate way to solve a problem like this.

We believe we can minimize the effect of rounding by using the law of cosines to find the magnitude.

  f = f1 +f2

  |f|² = |f1|² +|f2|² -2|f1|·|f2|·cos(180°-(170° -60°))

  |f|² = 200² +100² -2·200·100·cos(70°) ≈ 40,000 +10,000 -40,000·(0.34)

  |f|² ≈ 36,400

  |f| ≈ 190.79 . . . . . due to rounding cos(70°). Should be 190.576.

The angle of f can be computed from the law of sines. We choose to compute the angle of the summation triangle that is opposite f1. Call it ∠1. Then we have ...

  sin(∠1)/|f1| = sin(70°)/|f|

  ∠1 = arcsin(|f1|·sin(70°)/|f|) = arcsin(200·sin(70°)/190.79)

  ∠1 = arcsin(200·0.94/190.79) = arcsin(188/190.79) = arcsin(0.99)

  ∠1 = 81.89°

To find the resultant angle, this angle is subtracted from 170°:

  170° -81.89° = 88.11°

The direction of the resultant is 88.11° and its magnitude is 190.79 pounds.

Using appropriate end-of-calculation rounding, the resultant would be calculated as 190.56∠89.54° pounds.

If we compute the result by decomposing into horizontal and vertical components, we expect the effects of rounding to be far worse.

  200 cis 60° + 100 cis 170° = 200(cos(60°), sin(60°)) +100(cos(170°), sin(170°))

  = 200(0.50, 0.87) +100(-0.98, 0.17) = (100, 174) +(-98, 17) = (2, 191)

Then the magnitude is ...

  |f| = √(2² +191²) = √36485 ≈ 191.01

and the angle is ...

  ∠f = arctan(191/2) = arctan(95.50) = 89.40°

Calculated in this way, the resultant is ...

  direction: 89.40°, magnitude: 191.01 pounds

the surface area is 124cm²

step-by-step explanation:

length: 4cm

width: 9cm

height:   2cm

answer: 800

step-by-step explanation: