answer:1. Consider a rectangular parallelepiped labeled (A B C D A_1 B_1 C_1 D_1) with edges (A B = a), (A D = b), and (A A_1 = c). The diagonals intersect at point (O). We aim to find the angle between diagonals (A_1 C) and (B_1 D). The sides of the parallelepiped denote (overrightarrow{AB} = vec{a}), (overrightarrow{AD} = vec{b}), and (overrightarrow{AA_1} = vec{c}). 2. We need the vectors of the diagonals (A_1 C) and (B_1 D). [ overrightarrow{A_1 C} = overrightarrow{A_1 A} + overrightarrow{A B} + overrightarrow{B C} = -vec{c} + vec{a} + vec{b} ] [ overrightarrow{B_1 D} = overrightarrow{B_1 B} + overrightarrow{B A} + overrightarrow{A D} = -vec{c} -vec{a} + vec{b} ] 3. Using these vectors, apply the dot product to find the cosine of the angle between them. [ cosalpha = frac{overrightarrow{A_1 C} cdot overrightarrow{B_1 D}}{|overrightarrow{A_1 C}| |overrightarrow{B_1 D}|} ] 4. Compute (overrightarrow{A_1 C} cdot overrightarrow{B_1 D}): [ overrightarrow{A_1 C} cdot overrightarrow{B_1 D} = (-vec{c} + vec{a} + vec{b}) cdot (-vec{c} -vec{a} + vec{b}) = (-vec{c}) cdot (-vec{c}) + (-vec{c}) cdot (-vec{a}) + (-vec{c}) cdot vec{b} + vec{a} cdot (-vec{c}) + vec{a} cdot (-vec{a}) + vec{a} cdot vec{b} + vec{b} cdot (-vec{c}) + vec{b} cdot (-vec{a}) + vec{b} cdot vec{b} ] [ = c^2 + a^2 + b^2 - a^2 - b^2 - c^2 ] [ = -a^2 - b^2 - c^2 + 2b^2 = b^2 - a^2 - c^2 ] 5. Compute the magnitudes of (overrightarrow{A_1 C}) and (overrightarrow{B_1 D}): [ |overrightarrow{A_1 C}| = sqrt{a^2 + b^2 + c^2} ] [ |overrightarrow{B_1 D}| = sqrt{a^2 + b^2 + c^2} ] 6. Substituting these into the cosine formula: [ cosalpha = frac{b^2 - a^2 - c^2}{a^2 + b^2 + c^2} ] 9. The angle (alpha) is obtained by taking the arccosine: [ alpha = arccosleft(frac{b^2 - a^2 - c^2}{a^2 + b^2 + c^2}right) ] If (a^2 leq b^2 + c^2): [ boxed{alpha = arccosleft(frac{b^2 - a^2 - c^2}{a^2 + b^2 + c^2}right)} ] If (a^2 > b^2 + c^2): [ boxed{alpha = 180^circ - arccosleft(frac{b^2 - a^2 - c^2}{a^2 + b^2 + c^2}right)} ]

answer:The teacher initially had 34 worksheets to grade. After grading 7, she would have had 34 - 7 = 27 worksheets left to grade. However, she now has 63 worksheets to grade. This means that the number of worksheets turned in after she graded the first 7 is 63 - 27 = boxed{36} worksheets.

answer:First, let's find out how many words Isaiah can type in an hour. Since there are 60 minutes in an hour, and Isaiah can type 40 words per minute, we can calculate the number of words Isaiah can type in an hour by multiplying: Isaiah's words per hour = 40 words/minute * 60 minutes/hour = 2400 words/hour Now, we know that Isaiah can type 1200 more words than Micah in an hour. So, to find out how many words Micah can type in an hour, we subtract 1200 from Isaiah's total: Micah's words per hour = Isaiah's words per hour - 1200 words Micah's words per hour = 2400 words - 1200 words = 1200 words/hour Now, to find out how many words per minute Micah can type, we divide the number of words he can type in an hour by the number of minutes in an hour: Micah's words per minute = Micah's words per hour / 60 minutes/hour Micah's words per minute = 1200 words/hour / 60 minutes/hour = 20 words/minute Therefore, Micah can type boxed{20} words per minute.

answer:The ratio between a length on the model and a length on the actual Eiffel Tower is given as (6text{ inches} : 984text{ feet}). To find out how many feet of the Eiffel Tower one inch of the model represents, we divide the height of the Eiffel Tower by the height of the model: [ text{Ratio} = frac{984text{ feet}}{6text{ inches}} ] This simplifies to: [ text{Ratio} = frac{984}{6} = 164 text{ feet per inch} ] Thus, one inch on the model represents (boxed{164}) feet of the Eiffel Tower.