💡First, I need to determine the total number of each type of plant Shyne can grow from the seed packets she purchased.For eggplants:4 seed packets × 14 eggplants per packet = 56 eggplantsFor sunflowers:6 seed packets × 10 sunflowers per packet = 60 sunflowersFor tomatoes:5 seed packets × 16 tomatoes per packet = 80 tomatoesFor peas:7 seed packets × 20 peas per packet = 140 peasNext, I'll calculate how many of each plant Shyne can actually grow during the respective seasons, considering the space and growth season limitations.Eggplants in spring:56 eggplants × 70% = 39.2 eggplantsSince Shyne can't grow a fraction of an eggplant, we'll round down to 39 eggplants.Peas in spring:140 peas × 70% = 98 peasSunflowers in summer:60 sunflowers × 80% = 48 sunflowersTomatoes in summer:80 tomatoes × 80% = 64 tomatoesNow, I'll sum up the number of plants Shyne can grow in each season.Spring:39 eggplants + 98 peas = 137 plantsSummer:48 sunflowers + 64 tomatoes = 112 plantsFinally, I'll add the plants from both seasons to find the total number of plants Shyne can grow.Total plants:137 plants (spring) + 112 plants (summer) = 249 plants

💡Alright, so I have this problem here: Let ( n ) be the largest integer for which ( 14n ) has exactly 100 digits. I need to find the 68th digit of ( n ) when counting from the right. The options are (A) 1, (B) 2, (C) 4, (D) 5, and (E) 8.Hmm, okay. Let me start by understanding what it means for ( 14n ) to have exactly 100 digits. The number with exactly 100 digits would be between ( 10^{99} ) and ( 10^{100} - 1 ). So, ( 14n ) must be less than ( 10^{100} ), right?So, to find the largest ( n ), I need to solve for ( n ) in the inequality ( 14n < 10^{100} ). That means ( n < frac{10^{100}}{14} ). Let me compute that.Calculating ( frac{10^{100}}{14} ) is the same as ( frac{10}{14} times 10^{99} ). Simplifying ( frac{10}{14} ) gives ( frac{5}{7} ). So, ( n < frac{5}{7} times 10^{99} ).Now, ( frac{5}{7} ) is a repeating decimal. Let me recall, ( frac{1}{7} ) is approximately 0.142857142857..., so ( frac{5}{7} ) would be 0.714285714285... repeating every six digits. So, ( frac{5}{7} times 10^{99} ) would shift this decimal 99 places to the right.Wait, so ( frac{5}{7} times 10^{99} ) is essentially the number 714285714285... repeating up to 99 digits. Since 99 divided by 6 is 16.5, that means there are 16 full cycles of "714285" and then half of another cycle, which would be "714".So, the number ( n ) is just less than this, so it would be 714285714285...714. But since ( n ) has to be an integer, the largest integer less than ( frac{5}{7} times 10^{99} ) would be this repeating sequence without the decimal.Now, I need to find the 68th digit from the right in this number. Let me think about how to approach this.First, let's consider the structure of the number. It's a repeating pattern of "714285" every six digits. Since the number is 99 digits long, and the pattern repeats every six digits, there are 16 full repetitions (which take up 96 digits) and then three more digits: "714".So, the number looks like:- 16 repetitions of "714285" (96 digits)- Followed by "714" (3 digits)Therefore, the total number of digits is 99.Now, I need to find the 68th digit from the right. Let me count from the right, so the first digit from the right is the units place, the second is the tens place, and so on.Since the number is 99 digits long, the 68th digit from the right is the same as the (99 - 68 + 1) = 32nd digit from the left.Wait, let me verify that. If I have a 99-digit number, the first digit from the left is the 99th digit from the right, and the last digit from the left is the 1st digit from the right. So, the k-th digit from the right is the (99 - k + 1) digit from the left.So, for k = 68, it's (99 - 68 + 1) = 32nd digit from the left.So, I need to find the 32nd digit from the left in the number.Given the structure of the number, which is 16 repetitions of "714285" followed by "714", let's break it down.Each "714285" is 6 digits. So, 16 repetitions would cover 16 * 6 = 96 digits. Then, the last three digits are "714".So, the 32nd digit from the left would be within the first 16 repetitions.To find which part of the repetition it is, let's see how many full repetitions fit into 32 digits.Each repetition is 6 digits, so 32 divided by 6 is 5 with a remainder of 2.So, 5 full repetitions take up 30 digits, and the 31st and 32nd digits are the first two digits of the 6th repetition.Each repetition is "714285", so the first two digits are "7" and "1".Therefore, the 31st digit is "7", and the 32nd digit is "1".So, the 32nd digit from the left is "1", which means the 68th digit from the right is also "1".Wait, let me double-check that.If the 32nd digit from the left is "1", then counting from the right, it's the 68th digit. So, yes, that seems correct.Alternatively, maybe I can think of it as starting from the right. The last three digits are "714", so digits 97, 98, 99 from the left are "7", "1", "4".Then, before that, it's 16 repetitions of "714285". So, digits 1 to 96 are "714285" repeated 16 times.So, to find the 68th digit from the right, which is the 32nd digit from the left, it's within the first 96 digits.As calculated earlier, 32 divided by 6 is 5 with a remainder of 2. So, the 32nd digit is the 2nd digit of the 6th repetition.The 6th repetition starts at digit 31, so digit 31 is "7", digit 32 is "1".Therefore, the 68th digit from the right is "1".So, the answer should be (A) 1.But wait, let me make sure I didn't make a mistake in the counting.Another way to think about it is to list out the positions.Since the number is 99 digits long, and it's composed of "714285" repeated 16 times and then "714".So, positions 1-6: "714285"Positions 7-12: "714285"...Positions 91-96: "714285"Positions 97-99: "714"So, the 32nd digit from the left is in the 6th repetition (since 6*5=30, so 31-36 is the 6th repetition).Within the 6th repetition, the digits are "714285".So, the 31st digit is "7", 32nd is "1", 33rd is "4", 34th is "2", 35th is "8", 36th is "5".Therefore, yes, the 32nd digit is "1".So, counting from the right, the 68th digit is "1".I think that's solid. So, the answer is (A) 1.

💡Okay, so I have this problem here: I need to find the radius of a circle inscribed in a rhombus with diagonals of length 8 and 18. The options are given as A) 30 over root 97, B) 36 over root 97, C) 40 over root 97, and D) 42 over root 97. Hmm, alright, let me think about how to approach this.First, I remember that a rhombus is a type of quadrilateral where all sides are equal in length. Also, the diagonals of a rhombus bisect each other at right angles. So, if the diagonals are 8 and 18, they intersect each other at 90 degrees and cut each other exactly in half. That means each half of the diagonals would be 4 and 9 units long, right?Now, since the diagonals bisect each other at right angles, they form four right-angled triangles inside the rhombus. Each of these triangles has legs measuring 4 and 9 units. So, I can use the Pythagorean theorem to find the side length of the rhombus. Let me write that down:The side length (let's call it 'a') of the rhombus can be found using:[ a = sqrt{(d_1/2)^2 + (d_2/2)^2} ]Plugging in the values:[ a = sqrt{(8/2)^2 + (18/2)^2} = sqrt{4^2 + 9^2} = sqrt{16 + 81} = sqrt{97} ]So, each side of the rhombus is √97 units long.Next, I need to find the radius of the inscribed circle. I recall that the radius of an inscribed circle in a rhombus can be found using the formula:[ r = frac{text{Area of the rhombus}}{text{Perimeter of the rhombus}} ]But wait, let me make sure I have the correct formula. Alternatively, I remember that the area of a rhombus can also be calculated using the product of its diagonals divided by 2:[ text{Area} = frac{d_1 times d_2}{2} ]So, plugging in the given diagonals:[ text{Area} = frac{8 times 18}{2} = frac{144}{2} = 72 ]So, the area of the rhombus is 72 square units.Now, the perimeter of the rhombus is simply four times the side length, since all sides are equal:[ text{Perimeter} = 4 times a = 4 times sqrt{97} ]So, the perimeter is 4√97 units.Going back to the formula for the radius:[ r = frac{text{Area}}{text{Perimeter}} = frac{72}{4sqrt{97}} ]Simplifying that:[ r = frac{72}{4sqrt{97}} = frac{18}{sqrt{97}} ]Wait, that doesn't match any of the given options. Did I make a mistake somewhere?Let me double-check my steps. The area is definitely 72, and the perimeter is 4√97. So, 72 divided by 4√97 is indeed 18/√97. Hmm, but none of the options are 18/√97. The closest one is option A, which is 30/√97. Did I miscalculate something?Wait a second, maybe I used the wrong formula for the radius. Let me think again. I remember another formula for the radius of an inscribed circle in a rhombus, which is:[ r = frac{d_1 times d_2}{2 times sqrt{d_1^2 + d_2^2}} ]Let me try that. Plugging in the values:[ r = frac{8 times 18}{2 times sqrt{8^2 + 18^2}} = frac{144}{2 times sqrt{64 + 324}} = frac{144}{2 times sqrt{388}} ]Simplifying √388, which is √(4×97) = 2√97. So:[ r = frac{144}{2 times 2sqrt{97}} = frac{144}{4sqrt{97}} = frac{36}{sqrt{97}} ]Ah, that gives me 36/√97, which is option B. So, I must have used the wrong formula initially. The correct formula for the radius is indeed (d1×d2)/(2×√(d1² + d2²)). Therefore, the radius is 36/√97.Let me just verify this with another approach to be sure. The area of the rhombus is 72, and the semiperimeter would be half the perimeter, which is 2√97. Wait, no, the semiperimeter is half the perimeter, so 2√97. But actually, in the formula for the radius, it's area divided by semiperimeter for a general polygon, but for a rhombus, since it's a tangential quadrilateral, the radius is area divided by semiperimeter. Wait, no, the formula I used earlier was area divided by perimeter, which gave me 18/√97, but that was incorrect. The correct formula is area divided by semiperimeter. Let me check that.Semiperimeter (s) is half the perimeter:[ s = frac{4a}{2} = 2a = 2sqrt{97} ]So, radius r is area divided by semiperimeter:[ r = frac{text{Area}}{s} = frac{72}{2sqrt{97}} = frac{36}{sqrt{97}} ]Yes, that matches the previous result. So, the correct radius is indeed 36/√97, which is option B.I think I confused myself earlier by mixing up the formulas, but now I'm confident that option B is the correct answer.

💡Okay, so I have this function f(x) defined piecewise. For x greater than 0, it's f(x - 4), and for x less than or equal to 0, it's 2^x plus the integral from 0 to π/6 of cos(3x) dx. I need to find f(2016). Hmm, let's break this down.First, since 2016 is a positive number, I know I have to use the first case of the function, which is f(x - 4). So, f(2016) = f(2016 - 4) = f(2012). But wait, 2012 is still positive, so I guess I have to keep subtracting 4 until I get to a number that's less than or equal to 0. That way, I can use the second case of the function.Let me see, how many times do I need to subtract 4 from 2016 to get to a number less than or equal to 0? Well, 2016 divided by 4 is 504. So, if I subtract 4, 504 times, I'll get 2016 - 4*504 = 2016 - 2016 = 0. So, f(2016) = f(0). Okay, that makes sense.Now, I need to compute f(0). Since 0 is less than or equal to 0, I use the second case of the function: f(0) = 2^0 + the integral from 0 to π/6 of cos(3x) dx. Let's compute each part separately.First, 2^0 is straightforward. Any number to the power of 0 is 1, so 2^0 = 1.Next, I need to compute the integral of cos(3x) from 0 to π/6. I remember that the integral of cos(ax) dx is (1/a) sin(ax) + C, where C is the constant of integration. So, applying that here, the integral of cos(3x) dx should be (1/3) sin(3x) + C.Let's compute the definite integral from 0 to π/6. So, plugging in the upper limit first: (1/3) sin(3*(π/6)) = (1/3) sin(π/2). Sin(π/2) is 1, so this part is (1/3)*1 = 1/3.Now, plugging in the lower limit: (1/3) sin(3*0) = (1/3) sin(0) = 0. So, the integral from 0 to π/6 of cos(3x) dx is 1/3 - 0 = 1/3.Putting it all together, f(0) = 2^0 + 1/3 = 1 + 1/3 = 4/3. Therefore, f(2016) is also 4/3.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting with f(2016), since it's positive, I subtract 4 repeatedly until I get to 0. 2016 divided by 4 is indeed 504, so subtracting 4, 504 times gets me to 0. Then, computing f(0), which is 2^0 plus the integral. 2^0 is 1, and the integral of cos(3x) from 0 to π/6 is 1/3. Adding them together gives 4/3. Yeah, that seems correct.I think I got it right. So, the answer should be 4/3.**Final Answer**The value of ( f(2016) ) is boxed{dfrac{4}{3}}.

💡First, I'll denote the man's income in the first year as ( I ) and the portion he saves as ( S ). Therefore, his expenditure in the first year is ( I - S ).In the second year, his income increases by 20%, so his new income is ( I + 0.20I = 1.20I ).His savings in the second year increase by 100%, so his new savings are ( 2S ).His expenditure in the second year is his income minus his savings, which is ( 1.20I - 2S ).According to the problem, his total expenditure in 2 years is double his expenditure in the 1st year. So I can write the equation:[(I - S) + (1.20I - 2S) = 2(I - S)]Expanding and simplifying the equation:[I - S + 1.20I - 2S = 2I - 2S][2.20I - 3S = 2I - 2S]Now, I'll move the terms involving ( S ) to one side and the terms involving ( I ) to the other side:[2.20I - 2I = 3S - 2S][0.20I = S]To find the percentage of his income that he saved in the first year, I'll divide the savings by the income and multiply by 100:[text{Percentage saved} = left( frac{S}{I} right) times 100][text{Percentage saved} = left( frac{0.20I}{I} right) times 100][text{Percentage saved} = 0.20 times 100][text{Percentage saved} = 20%]

💡First, I need to determine the number of horses that "a" put into the pasture. I'll start by calculating the cost contributed by "b" and "c" based on the number of horses and the duration they were in the pasture.For "b":- Number of horses = 16- Duration = 9 months- Cost = 16 * 9 = 144For "c":- Number of horses = 18- Duration = 6 months- Cost = 18 * 6 = 108Next, I'll find the total cost contributed by "b" and "c":- Total cost by "b" and "c" = 144 + 108 = 252The total cost of the pasture is Rs. 435, so the cost contributed by "a" is:- Cost by "a" = 435 - 252 = 183Now, I'll calculate the number of horses "a" put in:- Let the number of horses "a" put in be "x"- Duration = 8 months- Cost by "a" = x * 8 = 183- Solving for x: x = 183 / 8 = 22.875Since the number of horses must be a whole number, I'll round 22.875 to the nearest whole number, which is 23.Therefore, "a" put in 23 horses.