answer:I fire up the engine of my worn, black sedan and hit the GPS with the address the stranger provided. The navigation system leads me out of the city, down a series of increasingly rural roads that wind through the dense, mist-shrouded woods on the outskirts of town. The farther I drive, the more isolated I feel - the streetlights and buildings giving way to dark, towering trees that seem to close in around me like sentinels. As I drive, my mind starts to wander. Who is this mysterious stranger, and what kind of case do they have for me? Is it a haunting? A possession? Something even weirder? I've dealt with my fair share of crackpots and pranksters over the years, but there was something about the stranger's voice that seemed...off. Urgent, almost desperate. I think back to some of my previous cases, trying to find any connections or similarities that might give me a clue about what I'm walking into. The vampire case from a few years ago? The haunted asylum? Nothing seems to quite fit, but I've got a nagging sense that I'm missing something. As the GPS leads me deeper into the woods, the trees grow closer together, casting long, ominous shadows across the road. I flip on my high beams, but they only seem to make the darkness more pronounced. I can feel my heart rate picking up, my senses going on high alert. Suddenly, my phone buzzes with an incoming text. It's from an unknown number - just a single sentence: Be careful, Detective. I frown, wondering who sent it and what they know. I try to call the number, but it goes straight to voicemail. The GPS announces my arrival at the destination, and I see a faint light up ahead. I slow down, my eyes scanning the surroundings. I'm at a large, old-fashioned mansion - the kind of place that looks like it's been abandoned for decades. But there's a light on in the front window, and I can sense...something. A presence, maybe. A feeling that I'm being watched. I take a deep breath, grab my notebook and flashlight, and step out into the darkness. Time to see what this mysterious stranger has in store for me.

answer:I approach the mansion cautiously, my eyes scanning the surroundings for any signs of movement or unusual activity. As I step out of the car, the crunch of gravel beneath my feet seems amplified in the stillness of the night. I pause for a moment, listening for any other sounds, but there's only an oppressive silence. I shine my flashlight around the perimeter of the property, taking in the overgrown gardens and the tangled vines that seem to be pulling the mansion back into the earth. The house itself looms above me, its facade a mix of Gothic and Victorian styles, with turrets and gargoyles reaching up towards the sky like skeletal fingers. As I move closer, I notice that the front door is slightly ajar. I can see a sliver of dim light coming from within, and I sense that I'm being invited inside. I push the door open slowly, calling out into the darkness, Hello? Is anyone here? My voice echoes off the walls, but there's no response. I step inside, my eyes adjusting to the dim light, and find myself in a grand foyer with a sweeping staircase curving up to the left. The air inside is musty and stale, filled with the scent of decay and rot. I notice that the walls are adorned with cobweb-covered portraits, and the floor is littered with dust and debris. It's clear that the mansion has been abandoned for a long time, but there are signs of recent occupation - a few faint footprints in the dust, a discarded coffee cup on a nearby table. I move deeper into the foyer, my flashlight casting eerie shadows on the walls. I can feel a presence watching me, a sense of unease that's growing with every step. Suddenly, I hear a faint noise coming from upstairs - a low, mournful sigh that sends a shiver down my spine. I hesitate for a moment, wondering if I should investigate further, but my curiosity gets the better of me. I start to creep up the stairs, my heart pounding in my chest, as I call out again, Hello? Is anyone here?

answer:1. The minor axis length is given as sqrt{5}. Using the relationship between the semi-minor axis and the semi-major axis, we have b = frac{sqrt{5}}{2}. 2. The eccentricity is given as e = frac{2}{3}, which implies frac{c}{a} = frac{2}{3}. Using the relationship between the semi-major axis, semi-minor axis, and the linear eccentricity, we have frac{sqrt{a^2 - frac{5}{4}}}{a} = frac{2}{3}. Solving for a, we get a = frac{3}{2}. 3. The perimeter of triangle ABF_2 is given by (|AF_1| + |AF_2|) + (|BF_1| + |BF_2|). By the definition of an ellipse, (|AF_1| + |AF_2|) = 2a and (|BF_1| + |BF_2|) = 2a. 4. Substituting a = frac{3}{2} into the expression, we get the perimeter of triangle ABF_2 as 2a + 2a = 4a = 6. Therefore, the perimeter of triangle ABF_2 is boxed{6}.

answer:To calculate the minimum number of widgets the company needs to sell each day to break even, we need to determine the company's total expenses and then divide that by the profit made per widget. The total expenses of the company can be calculated as follows: - Daily cost to produce 500 widgets = 500 x 5 = 2,500 - Daily overhead cost = 50,000 / 30 days = 1,666.67 - Total daily expenses = 2,500 + 1,666.67 = 4,166.67 The profit made per widget is the difference between the selling price and the cost to produce: - Profit per widget = 10 - 5 = 5 To break even, the company needs to sell enough widgets to cover its total expenses. So we can calculate the minimum number of widgets the company needs to sell each day as follows: - Minimum number of widgets = Total daily expenses / Profit per widget - Minimum number of widgets = 4,166.67 / 5 - Minimum number of widgets = 833.33 Therefore, the minimum number of widgets the company needs to sell each day to break even is 834. However, since we can't sell a fraction of a widget, the company would need to sell at least 835 widgets per day to break even.