Solve the problem.
-When a person bends at the waist with a straight back, the force, $\mathrm { F }$, exerted by the person's lower back muscles can be estimated with the formula $F = 2.89 w \sin \left( \theta + 90 ^ { \circ } \right)$, where $w$ is the person weight, and $\theta$ is the angle made by the person's torso relative to the horizontal. Suppose that a 200 -pound person bends at an angle of $30 ^ { \circ }$. Determine the force exerted by the person's back muscles, and estimate the angle that requires the back muscles to exert a force of 346 pounds. Round the force to the nearest pound, and the angle to the nearest tenth, if necessary.
A) 289 pounds; $47.3 ^ { \circ }$
B) 501 pounds; $53.2 ^ { \circ }$
C) 26 pounds; $53.2 ^ { \circ }$
D) 501 pounds; $30 ^ { \circ }$

Question

Quizplus · Accepted Answer

First, to find the force exerted by the person's back muscles when bending at a 30-degree angle, substitute $w = 200$ pounds and $\theta = 30^\circ$ into the formula: $F = 2.89 \times 200 \times \sin(30^\circ + 90^\circ)$. Since $\sin(120^\circ)$ is approximately 0.866, the force $F \approx 2.89 \times 200 \times 0.866 = 501$ pounds, rounded to the nearest pound. To find the angle that requires the back muscles to exert a force of 346 pounds, set $F = 346$ and solve for $\theta$. Rearranging the formula gives $\theta + 90^\circ = \arcsin\left(\frac{346}{2.89 \times 200}\right)$. Solving this equation gives an angle $\theta$ that, when rounded to the nearest tenth, matches the angle in choice B, indicating that both the force and the angle calculations align with this option.

Solve the Problem F\mathrm { F }F , Exerted by the Person's Lower Back Muscles Can Be

Solve the Problem $\mathrm { F }$ , Exerted by the Person's Lower Back Muscles Can Be