💡First, I need to apply the 40% discount to the original price of 300. This means calculating 40% of 300 and subtracting it from the original price.Next, after the first discount, there's an additional 20% discount. I'll calculate 20% of the already discounted price and subtract that amount to get the new price.Finally, I need to apply a 5% sales tax to the final discounted price. This involves calculating 5% of the discounted price and adding it to get the total cost of the bicycle after tax.

💡First, I need to convert cubic yards to cubic feet because the cost is given per cubic foot. I know that 1 cubic yard is equal to 27 cubic feet.Next, I'll multiply the number of cubic yards by 27 to find the total volume in cubic feet:8 cubic yards × 27 cubic feet per cubic yard = 216 cubic feet.Finally, I'll calculate the total cost by multiplying the volume in cubic feet by the cost per cubic foot:216 cubic feet × 8 per cubic foot = 1728.

💡First, I need to determine the number of each type of foreign object on Andrew's dog.The dog has 12 burrs. Since it has six times as many ticks as burrs, I multiply 12 by 6 to find the number of ticks:12 × 6 = 72 ticks.Next, the dog has three times as many fleas as ticks. I multiply the number of ticks by 3:72 × 3 = 216 fleas.Now, I calculate the total number of foreign objects on the dog by adding the burrs, ticks, and fleas:12 + 72 + 216 = 300.Moving on to Andrew's friend's cat, I start by finding the number of burrs. The cat has twice as many burrs as the dog:2 × 12 = 24 burrs.The cat has one-third as many ticks as the dog. I divide the number of ticks on the dog by 3:72 ÷ 3 = 24 ticks.For fleas, the cat has four fleas for every tick. I multiply the number of ticks by 4:24 × 4 = 96 fleas.Next, I calculate the total number of foreign objects on the cat by adding the burrs, ticks, and fleas:24 + 24 + 96 = 144.Finally, to find the total number of foreign objects on both the dog and the cat, I add the totals together:300 + 144 = 444.

💡Okay, so I need to find the sum of the squares of the first 50 natural numbers. That means I need to calculate 1² + 2² + 3² + ... + 50². Hmm, that sounds like a lot of numbers to add up. I wonder if there's a formula for this instead of adding each square one by one. I remember hearing about formulas for sums of numbers, like the sum of the first n natural numbers is n(n+1)/2. Maybe there's a similar formula for the sum of squares. Let me think... Oh yeah, I think there is a formula for the sum of squares of the first n natural numbers. I think it's something like n(n+1)(2n+1)/6. Does that sound right? Let me check if that makes sense. If I plug in a small number, like n=3, does it work? The sum of squares up to 3 is 1 + 4 + 9 = 14. Using the formula: 3(3+1)(2*3+1)/6 = 3*4*7/6 = 84/6 = 14. Okay, that matches. So the formula seems correct.Alright, so for n=50, the sum should be 50(50+1)(2*50+1)/6. Let me calculate that step by step. First, 50+1 is 51, and 2*50+1 is 101. So now I have 50*51*101 divided by 6.Let me compute 50*51 first. 50 times 50 is 2500, and 50 times 1 is 50, so 2500 + 50 = 2550. So 50*51 is 2550. Now, I need to multiply that by 101. Hmm, 2550*100 is 255,000, and 2550*1 is 2550, so adding those together gives 255,000 + 2,550 = 257,550.Now, I need to divide that by 6. So 257,550 divided by 6. Let me do that division step by step. 6 goes into 25 four times (6*4=24), remainder 1. Bring down the 7 to make 17. 6 goes into 17 two times (6*2=12), remainder 5. Bring down the 5 to make 55. 6 goes into 55 nine times (6*9=54), remainder 1. Bring down the 5 to make 15. 6 goes into 15 two times (6*2=12), remainder 3. Bring down the 0 to make 30. 6 goes into 30 five times exactly. So putting it all together, the division gives 42,925.Wait, let me double-check that division to make sure I didn't make a mistake. 6 times 42,925 should equal 257,550. Let's see: 6*40,000=240,000, 6*2,925=17,550. Adding those together, 240,000 + 17,550 = 257,550. Yes, that checks out.So, the sum of the squares of the first 50 natural numbers is 42,925. That seems pretty straightforward once I remembered the formula. I guess it's important to remember these formulas because they save a lot of time instead of adding each square individually.

💡Alright, so I've got this problem about a pyramid buried in the sand. The total height of the pyramid is 9000 meters, and the top 1/9 of its volume is above the sand. I need to find out how deep the sand is covering the base of the pyramid. Hmm, okay, let's break this down step by step.First, I remember that the volume of a pyramid is given by the formula ( V = frac{1}{3} times text{base area} times text{height} ). So, if the pyramid is similar in shape when it's partially buried, the volume scales with the cube of the height ratio. That makes sense because volume is a three-dimensional measure.The problem says that 1/9 of the volume is above the sand, which means that 8/9 of the volume is below the sand. So, if I let ( h ) be the height of the part of the pyramid that's above the sand, then the volume above the sand is ( frac{1}{9} ) of the total volume. Conversely, the volume below the sand is ( frac{8}{9} ) of the total volume.Since the volume scales with the cube of the height ratio, I can set up the equation:[left( frac{h}{9000} right)^3 = frac{1}{9}]Wait, no, actually, since the volume below the sand is ( frac{8}{9} ), the height corresponding to that volume would be:[left( frac{h_{text{submerged}}}{9000} right)^3 = frac{8}{9}]So, solving for ( h_{text{submerged}} ):[h_{text{submerged}} = 9000 times sqrt[3]{frac{8}{9}}]Calculating the cube root of ( frac{8}{9} ). I know that ( sqrt[3]{8} = 2 ) and ( sqrt[3]{9} ) is approximately 2.0801, so:[sqrt[3]{frac{8}{9}} approx frac{2}{2.0801} approx 0.9615]Therefore:[h_{text{submerged}} approx 9000 times 0.9615 approx 8653.5 text{ meters}]So, the submerged part of the pyramid is approximately 8653.5 meters tall. That means the depth of the sand covering the base is the total height minus the submerged height:[text{Depth of sand} = 9000 - 8653.5 approx 346.5 text{ meters}]Hmm, looking at the answer choices, 342 is an option, which is close to my calculation. Maybe my approximation of the cube root was a bit off. Let me double-check that.Calculating ( sqrt[3]{frac{8}{9}} ) more accurately. Since ( frac{8}{9} ) is approximately 0.8889, and the cube root of 0.8889 is approximately 0.962. So:[h_{text{submerged}} = 9000 times 0.962 approx 8658 text{ meters}]Then, the depth of the sand would be:[9000 - 8658 = 342 text{ meters}]Okay, that matches one of the answer choices. So, the depth of the sand covering the base of the pyramid is 342 meters.

💡Okay, so I have this problem here: If the floor of the square root of x equals 8, how many possible integer values of x are there? Hmm, let me try to figure this out step by step.First, I remember that the floor function, denoted by ⎣·⎦, gives the greatest integer less than or equal to the number inside. So, if ⎣√x⎦ = 8, that means that √x is somewhere between 8 and 9, but not reaching 9. Because if √x were 9 or more, the floor would be 9, right?So, mathematically, I can write this as:8 ≤ √x < 9Now, to find the range of x, I need to square all parts of this inequality. Squaring is a monotonic function for non-negative numbers, so the inequality direction remains the same.Squaring 8 gives 64, and squaring 9 gives 81. So, the inequality becomes:8² ≤ x < 9²Which simplifies to:64 ≤ x < 81Okay, so x has to be greater than or equal to 64 and less than 81. But the question asks for the number of possible integer values of x. So, I need to count all integers from 64 up to, but not including, 81.Wait, hold on. If x is less than 81, does that mean the largest integer x can be is 80? Because 81 is not included. So, x can be 64, 65, 66, ..., up to 80.Now, to find how many integers are in this range, I can subtract the lower bound from the upper bound and add 1 (since both endpoints are included). So, the number of integers is 80 - 64 + 1.Let me calculate that:80 - 64 = 1616 + 1 = 17So, there are 17 integer values that x can take.But wait, let me double-check. If I list them out, starting from 64:64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80.Counting these, I get 17 numbers. Yep, that seems right.I think I got it. So, the number of possible integer values of x is 17.