1. **Problem Statement:** Juana and Diego pull a stubborn puppy with forces of 23 lb and 27 lb at angles 18° and 15°, respectively. Find the force the puppy exerts to hold them at a standstill, i.e., the resultant force. 2. **Formula and Rules:** The resultant force vector \( \mathbf{R} \) is the vector sum of the two forces: $$\mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2$$ where each force vector can be expressed in component form: $$\mathbf{F} = F \langle \cos \theta, \sin \theta \rangle$$ Angles are measured from the positive x-axis (assumed horizontal). 3. **Calculate components of each force:** - For \( \mathbf{F}_1 \) with magnitude 23 lb and angle 18°: $$F_{1x} = 23 \cos 18^\circ$$ $$F_{1y} = 23 \sin 18^\circ$$ - For \( \mathbf{F}_2 \) with magnitude 27 lb and angle 15°: $$F_{2x} = 27 \cos 15^\circ$$ $$F_{2y} = 27 \sin 15^\circ$$ 4. **Evaluate components numerically:** Using approximate values: $$\cos 18^\circ \approx 0.9511, \sin 18^\circ \approx 0.3090$$ $$\cos 15^\circ \approx 0.9659, \sin 15^\circ \approx 0.2588$$ Calculate: $$F_{1x} = 23 \times 0.9511 = 21.8753$$ $$F_{1y} = 23 \times 0.3090 = 7.1070$$ $$F_{2x} = 27 \times 0.9659 = 26.0793$$ $$F_{2y} = 27 \times 0.2588 = 6.9876$$ 5. **Sum components to find resultant:** $$R_x = F_{1x} + F_{2x} = 21.8753 + 26.0793 = 47.9546$$ $$R_y = F_{1y} + F_{2y} = 7.1070 + 6.9876 = 14.0946$$ 6. **Interpretation:** The puppy pulls with a force equal in magnitude and opposite in direction to \( \mathbf{R} \) to hold the children at a standstill. **Final answer:** $$\boxed{\mathbf{R} = \langle 47.95, 14.09 \rangle}$$ This is the component form of the force the puppy exerts.