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question:Given vectors overrightarrow{a} = (2, 3) and overrightarrow{b} = (2, 1), the projection of overrightarrow{a} in the direction of overrightarrow{b} is equal to ______.
answer:According to the definition of projection, the projection of overrightarrow{a} in the direction of overrightarrow{b} is |overrightarrow{a}|coslangleoverrightarrow{a}, overrightarrow{b}rangle = frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{b}|} = -frac{sqrt{5}}{5}. Therefore, the answer is -frac{sqrt{5}}{5}. By applying the formula |overrightarrow{a}|coslangleoverrightarrow{a}, overrightarrow{b}rangle = frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{b}|} based on the definition of projection, we can solve the problem. This question mainly tests the understanding of the definition of vector projection and requires proficient application of the formula. Thus, the final answer is boxed{-frac{sqrt{5}}{5}}.
question:5 years ago , Jay was seven years older than Mary . In five years , Jay will be twice as old as Mary . How old is Mary now ?
answer:Let's use algebra to solve this problem. Let ( M ) represent Mary's current age and ( J ) represent Jay's current age. From the first statement, "5 years ago, Jay was seven years older than Mary," we can write the following equation: [ J - 5 = (M - 5) + 7 ] Simplifying this, we get: [ J - 5 = M + 2 ] [ J = M + 7 ] (Equation 1) From the second statement, "In five years, Jay will be twice as old as Mary," we can write the following equation: [ J + 5 = 2(M + 5) ] Simplifying this, we get: [ J + 5 = 2M + 10 ] [ J = 2M + 5 ] (Equation 2) Now we have two equations with two variables: 1. ( J = M + 7 ) 2. ( J = 2M + 5 ) We can set these two equations equal to each other since they both equal ( J ): [ M + 7 = 2M + 5 ] Now, let's solve for ( M ): [ 7 - 5 = 2M - M ] [ 2 = M ] So, Mary is currently 2 years old. To check our work, we can plug this value back into the equations: 1. If Mary is 2, then Jay is ( 2 + 7 = 9 ) years old now. 2. In five years, Mary will be ( 2 + 5 = 7 ) years old, and Jay will be ( 9 + 5 = 14 ) years old, which is indeed twice Mary's age in five years. Therefore, Mary is currently boxed{2} years old.
question:A square with sides of length 8 inches is given. If ( P ) is a point within the square such that segments (overline{PA}), (overline{PB}), and (overline{PC}) are equal in length and segment (overline{PD}) is perpendicular to segment (overline{AB}), find the area, in square inches, of triangle ( APB ). Assume point ( A ) is the bottomleft corner, ( B ) is the bottomright corner, ( C ) is the topright corner, and ( D ) is the topleft corner of the square.
answer:We let ( x = ) the lengths of segments (overline{PA}), (overline{PB}), and (overline{PC}). Extend line segment (overline{PD}) to intersect (overline{AB}) at point ( E ), making ( overline{DE} ) the perpendicular bisector. Thus, ( AE = EB = 4 ) inches (half the side of the square). Since ( triangle AEP ) is a right triangle and ( overline{PD} ) being perpendicular to (overline{AB}), we apply the Pythagorean Theorem: [ AE^2 + PE^2 = PA^2 ] Substituting known values, we get: [ 4^2 + (8-x)^2 = x^2 ] [ 16 + 64 - 16x + x^2 = x^2 ] [ 16x = 80 ] [ x = 5 ] Thus, the height of triangle ( APB ) from point ( P ) to base ( AB ) is ( 8 - x = 3 ) inches. The base of ( triangle APB ) is (8) inches, the length of side ( AB ). Therefore, the area is: [ text{Area} = frac{1}{2} times text{base} times text{height} = frac{1}{2} times 8 times 3 = boxed{12} text{ square inches} ]
question:Mr. Grey is purchasing gifts for his family. His shopping list includes: 1. 3 polo shirts for 26 each 2. 2 necklaces for 83 each 3. 1 computer game for 90 4. 4 pairs of socks for 7 each 5. 3 books for 15 each 6. 2 designer scarves at 22 each 7. 5 decorative mugs for 8 each 8. 1 pair of sneakers for 65. All of Mr. Grey's purchases are subjected to a 6.5% sales tax. Since he used his credit card for his shopping, he received a 12 rebate. Furthermore, Mr. Grey also has coupons for a 10% discount on books and a 15% discount on the pair of sneakers. What is the total cost of the gifts Mr. Grey purchased after the sales tax, discounts from coupons and credit card rebate?
answer:First, let's calculate the total cost of each item before any discounts or taxes: 1. Polo shirts: 3 * 26 = 78 2. Necklaces: 2 * 83 = 166 3. Computer game: 1 * 90 = 90 4. Socks: 4 * 7 = 28 5. Books: 3 * 15 = 45 6. Designer scarves: 2 * 22 = 44 7. Decorative mugs: 5 * 8 = 40 8. Sneakers: 1 * 65 = 65 Now, let's calculate the total cost before taxes and discounts: Total cost = 78 + 166 + 90 + 28 + 45 + 44 + 40 + 65 = 556 Next, we apply the discounts on books and sneakers: Discount on books = 10% of 45 = 0.10 * 45 = 4.50 Discount on sneakers = 15% of 65 = 0.15 * 65 = 9.75 Now, we subtract the discounts from the total cost: Total cost after discounts = 556 - 4.50 - 9.75 = 541.75 Now, we calculate the sales tax: Sales tax = 6.5% of 541.75 = 0.065 * 541.75 = 35.21375 We add the sales tax to the total cost after discounts: Total cost after discounts and tax = 541.75 + 35.21375 = 576.96375 Finally, we subtract the credit card rebate: Total cost after discounts, tax, and rebate = 576.96375 - 12 = 564.96375 Rounding to the nearest cent, the total cost of the gifts Mr. Grey purchased is boxed{564.96} .