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Andrew is pulling a sled with a force of 275 newtons by holding a rope at a 58^(@) angle. His brother Austin is pushing the sled with a force of 320 newtons. Determine the magnitude and direction of the resultant force on the sled. Round all angles to the nearest tenth and sides to the nearest hundredths. 4 pts

Andrew is pulling a sled with a force of 275275 newtons by holding a rope at a 58∘ 58^{\circ} angle. His brother Austin is pushing the sled with a force of 320320 newtons. Determine the magnitude and direction of the resultant force on the sled. Round all angles to the nearest tenth and sides to the nearest hundredths.

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Q. Andrew is pulling a sled with a force of 275275 newtons by holding a rope at a 58∘ 58^{\circ} angle. His brother Austin is pushing the sled with a force of 320320 newtons. Determine the magnitude and direction of the resultant force on the sled. Round all angles to the nearest tenth and sides to the nearest hundredths.
  1. Identify components of forces: Identify the components of the forces.\newlineAndrew is pulling the sled at an angle, so his force has both horizontal and vertical components. Austin is pushing horizontally, so his force has only a horizontal component.\newlineTo find the horizontal component of Andrew's force, we use the cosine of the angle:\newlineHorizontal component of Andrew's force = 275×cos⁡(58∘)275 \times \cos(58^\circ)\newlineTo find the vertical component of Andrew's force, we use the sine of the angle:\newlineVertical component of Andrew's force = 275×sin⁡(58∘)275 \times \sin(58^\circ)\newlineAustin's force is entirely horizontal, so it has no vertical component.
  2. Calculate components of Andrew's force: Calculate the horizontal and vertical components of Andrew's force.\newlineHorizontal component of Andrew's force = 275×cos⁡(58°)≈275×0.52992≈145.73275 \times \cos(58°) \approx 275 \times 0.52992 \approx 145.73 newtons\newlineVertical component of Andrew's force = 275×sin⁡(58°)≈275×0.84805≈233.21275 \times \sin(58°) \approx 275 \times 0.84805 \approx 233.21 newtons
  3. Sum horizontal forces: Sum the horizontal forces.\newlineTotal horizontal force = Horizontal component of Andrew's force + Austin's force\newlineTotal horizontal force = 145.73145.73 newtons + 320320 newtons = 465.73465.73 newtons
  4. Total vertical force: Since Austin is only pushing horizontally, the total vertical force is just the vertical component of Andrew's force.\newlineTotal vertical force =Vertical component of Andrew’s force=233.21 newtons= \text{Vertical component of Andrew's force} = 233.21 \text{ newtons}
  5. Calculate resultant force magnitude: Calculate the magnitude of the resultant force using the Pythagorean theorem.\newlineMagnitude of the resultant force = (Total horizontal force2+Total vertical force2)\sqrt{(\text{Total horizontal force}^2 + \text{Total vertical force}^2)}\newlineMagnitude of the resultant force = (465.732+233.212)\sqrt{(465.73^2 + 233.21^2)}\newlineMagnitude of the resultant force = (217105.53+54391.36)\sqrt{(217105.53 + 54391.36)}\newlineMagnitude of the resultant force = (271496.89)\sqrt{(271496.89)}\newlineMagnitude of the resultant force ≈521.05\approx 521.05 newtons
  6. Calculate resultant force direction: Calculate the direction of the resultant force using the tangent function.\newlineDirection of the resultant force = arctan⁡(Total vertical forceTotal horizontal force)\arctan\left(\frac{\text{Total vertical force}}{\text{Total horizontal force}}\right)\newlineDirection of the resultant force = arctan⁡(233.21465.73)\arctan\left(\frac{233.21}{465.73}\right)\newlineDirection of the resultant force ≈arctan⁡(0.5005)\approx \arctan(0.5005)\newlineDirection of the resultant force ≈26.6∘\approx 26.6^\circ (rounded to the nearest tenth)

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