💡First, I need to determine the largest possible circle that can fit within a rectangle that is 5 cm long and 4 cm wide.To do this, I will consider the dimensions of the rectangle. The largest circle that can fit inside the rectangle will have a diameter equal to the shorter side of the rectangle to ensure it fits within both the length and the width.Next, I will calculate the radius of the circle by dividing the diameter by 2.Once I have the radius, I can calculate the area of the circle using the formula ( pi r^2 ).After finding the area of the circle, I will calculate the area of the rectangle using the formula ( text{length} times text{width} ).Finally, I will determine what percentage of the rectangle's area is occupied by the circle by dividing the area of the circle by the area of the rectangle and multiplying by 100.

💡First, I need to determine how many berries all 5 birds consume in one day. Given that they eat a total of 140 berries over 4 days, I can calculate the daily consumption by dividing 140 by 4.140 berries ÷ 4 days = 35 berries per day for all 5 birds.Next, to find out how many berries one bird eats in a day, I divide the total daily consumption by the number of birds.35 berries per day ÷ 5 birds = 7 berries per bird per day.Therefore, each bird eats 7 berries in a day.

💡Okay, so I need to find the angle of inclination of the tangent line to the curve ( y = x^3 - 2x + m ) at ( x = 1 ). Hmm, I'm a bit rusty on this, but I think it involves calculus, specifically derivatives. Let me try to recall.First, I remember that the derivative of a function gives the slope of the tangent line at any point on the curve. So, if I can find the derivative of ( y ) with respect to ( x ), that should give me the slope at any ( x ), including ( x = 1 ).Alright, let's find the derivative. The function is ( y = x^3 - 2x + m ). The derivative of ( x^3 ) is ( 3x^2 ), the derivative of ( -2x ) is ( -2 ), and the derivative of a constant like ( m ) is zero. So putting that together, the derivative ( y' ) should be ( 3x^2 - 2 ). Wait, let me double-check that. Yeah, the power rule says that the derivative of ( x^n ) is ( nx^{n-1} ), so ( x^3 ) becomes ( 3x^2 ), and ( -2x ) becomes ( -2 ). Constants disappear because their slope is zero. Okay, that seems right.Now, I need to find the slope at ( x = 1 ). So, plug ( x = 1 ) into the derivative. That would be ( 3(1)^2 - 2 ). Calculating that, ( 1^2 ) is 1, multiplied by 3 gives 3, and then subtract 2, so 3 - 2 is 1. So the slope ( m ) of the tangent line at ( x = 1 ) is 1.Wait, hold on, the problem mentions ( m ) in the original function. Is that the same ( m ) as the slope? No, I think in the function ( y = x^3 - 2x + m ), ( m ) is just a constant term, like the y-intercept. So when I took the derivative, that constant went away, which is why the derivative is ( 3x^2 - 2 ). So the slope at ( x = 1 ) is indeed 1, regardless of the value of ( m ). That makes sense because the constant term doesn't affect the slope.Okay, so now I have the slope of the tangent line at ( x = 1 ) is 1. The next step is to find the angle of inclination. I think the angle of inclination is the angle that the tangent line makes with the positive direction of the x-axis. So, if I can find that angle, that should be the answer.I remember that the slope of a line is equal to the tangent of the angle it makes with the x-axis. So, if the slope is ( m ), then ( m = tan(theta) ), where ( theta ) is the angle of inclination. In this case, the slope is 1, so ( 1 = tan(theta) ).Now, I need to find ( theta ) such that ( tan(theta) = 1 ). I recall that ( tan(45^circ) = 1 ), because in a 45-45-90 triangle, the legs are equal, so the ratio of opposite over adjacent is 1. So, ( theta = 45^circ ).Wait, is there any possibility of another angle where the tangent is 1? Well, tangent has a period of 180 degrees, so technically, ( theta = 45^circ + k times 180^circ ) for any integer ( k ). But since we're talking about the angle of inclination, which is the smallest positive angle between the line and the x-axis, it should be between 0 and 180 degrees. So, the smallest positive angle where tangent is 1 is 45 degrees.Therefore, the angle of inclination is 45 degrees.Let me just recap to make sure I didn't miss anything. I found the derivative of the function, which gave me the slope at any point ( x ). Plugging in ( x = 1 ) gave me a slope of 1. Then, since the slope is equal to the tangent of the angle of inclination, I set ( tan(theta) = 1 ) and solved for ( theta ), which is 45 degrees. I think that's all there is to it. The constant ( m ) in the original function didn't affect the slope, so it doesn't matter what ( m ) is; the angle of inclination will always be 45 degrees at ( x = 1 ) for this curve.**Final Answer**The angle of inclination of the tangent line to the curve at ( x = 1 ) is boxed{45^circ}.

💡Okay, so I have this problem here. It says, given the set A equals all y such that y is 2 to the power of x, where x is a real number. Then, it's asking for the complement of A in the real numbers, which is written as complement_R A. The options are A: empty set, B: (-∞, 0], C: (0, +∞), and D: R.Alright, let me break this down. First, I need to understand what set A actually is. The set A is defined as all real numbers y where y equals 2 raised to some real number x. So, mathematically, A = { y | y = 2^x, x ∈ R }.Hmm, okay. So, 2^x is an exponential function. I remember that exponential functions like 2^x have certain properties. For example, 2^x is always positive because any positive number raised to any real power is still positive. So, no matter what x is, whether it's positive, negative, or zero, 2^x will always be greater than zero. That means every element y in set A is a positive real number.So, if I think about it, set A is just all positive real numbers. In interval notation, that would be (0, +∞). So, A = (0, +∞).Now, the question is asking for the complement of A in the real numbers. The complement of a set A, denoted as complement_R A, is all the real numbers that are not in A. So, if A is (0, +∞), then the complement would be all real numbers less than or equal to zero, right?Let me write that down. If A is (0, +∞), then complement_R A is (-∞, 0]. Because any real number that's not positive must be zero or negative. So, that includes all negative numbers and zero.Looking back at the options, option B is (-∞, 0], which matches what I just figured out. Option A is the empty set, which doesn't make sense because there are definitely real numbers not in A. Option C is (0, +∞), which is actually set A itself, not the complement. Option D is all real numbers, which would mean the complement is everything, but that's not true because A is a subset of the real numbers, so its complement can't be everything.Wait, just to make sure I didn't make a mistake. Let me double-check. If A is all positive real numbers, then the complement should be all non-positive real numbers, which is zero and everything less than zero. So, yeah, that's (-∞, 0]. So, option B is correct.I think that's solid. I don't see any mistakes in my reasoning. So, I feel confident that the answer is B.**Final Answer**boxed{B}

💡First, I need to understand the operation ( spadesuit ) defined as ( x spadesuit y = (x + y)(x - y) ). This operation essentially calculates the difference of squares, which simplifies to ( x^2 - y^2 ).To solve ( 3 spadesuit (5 spadesuit 6) ), I'll start by evaluating the inner operation ( 5 spadesuit 6 ).Calculating ( 5 spadesuit 6 ):[5 spadesuit 6 = (5 + 6)(5 - 6) = 11 times (-1) = -11]Now, substitute ( -11 ) back into the original expression:[3 spadesuit (5 spadesuit 6) = 3 spadesuit (-11)]Next, evaluate ( 3 spadesuit (-11) ):[3 spadesuit (-11) = (3 + (-11))(3 - (-11)) = (-8) times 14 = -112]Therefore, the final result of ( 3 spadesuit (5 spadesuit 6) ) is (-112).

💡First, I need to determine the certain number when 40% of it equals 160. To do this, I'll set up the equation 0.40 multiplied by the number equals 160.Next, I'll solve for the number by dividing both sides of the equation by 0.40. This gives me the number as 400.Now that I have the number, I can find 60% of it by multiplying 400 by 0.60.Finally, calculating 0.60 times 400 results in 240. Therefore, 60% of the number is 240.