answer:Let's denote the number of miles the car can drive in one hour as ( x ). According to the information given, the car drives for 5 hours and then needs 1 hour to cool down. This cycle (5 hours of driving + 1 hour of cooling down) takes a total of 6 hours. In 13 hours, the car would complete 2 full cycles of this 6-hour pattern (5 hours driving + 1 hour cooling down), which would take 12 hours, and then it would drive for an additional 1 hour (since 13 - 12 = 1). So, the total driving time in 13 hours is the driving time in the two full cycles (which is 5 hours driving x 2 cycles = 10 hours) plus the additional 1 hour of driving, giving us a total of 11 hours of driving. We are told that the car can drive 88 miles in 13 hours. Since the car is actually driving for 11 of those hours, we can set up the following equation: ( 11x = 88 ) Now, we solve for ( x ) to find out how many miles the car can drive in one hour: ( x = frac{88}{11} ) ( x = 8 ) Therefore, the car can drive boxed{8} miles in one hour.

answer:Given uv = 20 - 15i, we calculate the magnitude: [ |uv| = |20 - 15i| = sqrt{20^2 + (-15)^2} = sqrt{400 + 225} = sqrt{625} = 25. ] Since |uv| = |u||v|, we have: [ |u||v| = 25. ] Given |u| = 5, solving for |v|: [ 5|v| = 25 ] [ |v| = frac{25}{5} = 5. ] Thus, |v| = boxed{5}.

answer:Consider the quadrilateral (ABCD), where (angle B = angle D = 60^circ). 1. **Constructing Point (P)**: - Let (M) be the midpoint of (AD). - Draw a line through (M) parallel to (CD), intersecting (BC) at point (P). 2. **Examining the Properties of ( triangle ABP )**: - Given ( angle B = 60^circ ) and (angle D = 60^circ), which implies (BCD) is a triangle with (angle BCD = 120^circ). 3. **Constructing the Isosceles Triangle ( triangle MBX' )**: - Construct an equilateral triangle ( triangle MBX' ) such that point (X') lies on the same side of (BM) as (X). 4. **Condition for (AB = BP)**: - To prove (AB = BP), it implies ( triangle ABP ) must also form an isosceles triangle. Consequently, ( angle ABP = 60^circ ). 5. **Calculation of Angles**: - Therefore, ( angle BAP = angle BMP = 60^circ ). - We then have: [ angle ABM = angle PBX' = 60^circ ] - Noting ( BM = BX' ) (because ( triangle MBX' ) is equilateral), if ( AB = BP ), the triangles ( triangle BAM ) and ( triangle BPX' ) are congruent (by SAS congruence criterion). 6. **Congruence Implies Equal Segments**: - Since ( triangle BAM cong triangle BPX' ): [ Rightarrow PX' = AM = MD ] - Given (MD' = MD) when drawn through (M), the point (D') overlaps with (D), and thus, point (X') lies on line (CD). 7. **Conclusion**: - Therefore, the overlapping of points (X') and (X) indicates the necessary condition is met. - Finally, the condition ( angle MXB = 60^circ) implies: [ boxed{angle MXB = 60^circ text{ is a necessary and sufficient condition for } AB = BP} ] This confirms the precise condition needed for (AB = BP) is indeed ( angle MXB = 60^circ ).

answer:1. **Identify the problem and given parameters:** Let's denote the given problem as finding a pair of numbers that satisfies certain conditions, which are not explicitly mentioned in the reference. Assume one of the numbers is found correctly but there is an arithmetic error in the last step of the solution process. 2. **Translate the verbal problem into a mathematical formulation:** While the exact problem is not specified here, let's suppose we need to find two numbers x and y such that they satisfy a system of equations. For instance, let the equations be: [ x + y = S quad text{(sum of the numbers)} ] and [ xy = P quad text{(product of the numbers)} ] 3. **Solve one of the equations in terms of a single variable:** From the first equation, express y in terms of x: [ y = S - x ] 4. **Substitute this expression into the second equation:** Replace y in the product equation: [ x(S - x) = P ] [ xS - x^2 = P ] [ x^2 - Sx + P = 0 ] 5. **Solve the quadratic equation:** Using the quadratic formula x = frac{-b pm sqrt{b^2 - 4ac}}{2a} where a = 1, b = -S, c = P, we get: [ x = frac{S pm sqrt{S^2 - 4P}}{2} ] 6. **Determine both x and y:** After finding x, find y using y = S - x. We obtain two possible pairs of (x, y): [ x_1 = frac{S + sqrt{S^2 - 4P}}{2}, quad y_1 = frac{S - sqrt{S^2 - 4P}}{2} ] [ x_2 = frac{S - sqrt{S^2 - 4P}}{2}, quad y_2 = frac{S + sqrt{S^2 - 4P}}{2} ] 7. **Check and conclude:** Given that one of the numbers is correctly found, let's assume the error was in the arithmetic calculation or misunderstanding of the sign in the quadratic formula. Check the discriminant to verify if both solutions are valid. [ S^2 - 4P geq 0 ] This ensures real solutions for the quadratic equation. If only one solution satisfies the problem constraints (which might be context-specific), explicitly deduce why the solution is unique. 8. **Final correct answer:** Given the conditions explained in the problem statement, the corrected and verified final answer pair for the numbers would be: [ boxed{(x, y) = left( frac{S + sqrt{S^2 - 4P}}{2}, frac{S - sqrt{S^2 - 4P}}{2} right) text{ or } left( frac{S - sqrt{S^2 - 4P}}{2}, frac{S + sqrt{S^2 - 4P}}{2} right)} ]