💡First, I'll assume the original price of the shirt is 100 to simplify the calculations.Next, the shirt is sold at 90% of its original price. This means the sale price is:100 multiplied by 90%, which equals 90.One week later, the sale price is reduced by an additional 10%. To find the final price, I'll calculate 10% of 90 and subtract it from 90:10% of 90 is 9, so the final price is 90 minus 9, which equals 81.Finally, to determine what percentage 81 is of the original 100 price, I'll divide 81 by 100 and multiply by 100%:(81 / 100) * 100% equals 81%.Therefore, the final price is 81% of the original price.

💡Okay, so I have this problem about a tetrahedron (ABCD) where the edges (AB = BC = CA) and (DA = DB = DC). I need to prove that (cos angle AOB + cos angle AOD geq -frac{2}{3}), where (O) is the center of the circumscribed sphere.First, let me try to visualize the tetrahedron. Since (AB = BC = CA), the base triangle (ABC) is equilateral. Similarly, (DA = DB = DC) suggests that point (D) is equidistant from (A), (B), and (C). So, (D) must lie somewhere along the perpendicular line from the centroid of triangle (ABC).Given that (O) is the circumcenter, it should be equidistant from all four vertices (A), (B), (C), and (D). In a regular tetrahedron, the circumcenter coincides with the centroid, but this tetrahedron isn't necessarily regular because the edges from (D) to the base might not be equal to the edges of the base. So, (O) might not coincide with the centroid.Let me recall that in a tetrahedron, the circumcenter is the point equidistant from all four vertices, so it's the intersection point of the perpendicular bisectors of the edges. Since (ABC) is an equilateral triangle, its circumcenter is also its centroid. Let me denote the centroid of triangle (ABC) as (G). Then, (G) is the point where the medians intersect, and it's also the circumcenter of triangle (ABC).Now, since (DA = DB = DC), point (D) lies on the perpendicular line from (G). Let me denote the line (DG) as the axis of the tetrahedron. The circumcenter (O) must lie somewhere along this axis because it needs to be equidistant from (A), (B), (C), and (D).Let me consider the coordinates to make things clearer. Maybe placing the tetrahedron in a coordinate system will help. Let me set point (G) at the origin ((0, 0, 0)). Since (ABC) is an equilateral triangle, I can place the points as follows:- Let (A = (1, 0, 0))- (B = (-frac{1}{2}, frac{sqrt{3}}{2}, 0))- (C = (-frac{1}{2}, -frac{sqrt{3}}{2}, 0))This way, triangle (ABC) is equilateral with side length (sqrt{3}) (distance between (A) and (B) is (sqrt{(1 + frac{1}{2})^2 + (0 - frac{sqrt{3}}{2})^2} = sqrt{frac{9}{4} + frac{3}{4}} = sqrt{3})).Point (D) lies along the z-axis, so let me denote (D = (0, 0, h)) for some height (h). Now, since (DA = DB = DC), the distance from (D) to each of (A), (B), and (C) is the same. Let me compute (DA):(DA = sqrt{(1 - 0)^2 + (0 - 0)^2 + (0 - h)^2} = sqrt{1 + h^2}).Similarly, (DB = sqrt{(-frac{1}{2} - 0)^2 + (frac{sqrt{3}}{2} - 0)^2 + (0 - h)^2} = sqrt{frac{1}{4} + frac{3}{4} + h^2} = sqrt{1 + h^2}).Same for (DC), so that's consistent.Now, the circumradius (R) of the tetrahedron can be found by finding the radius of the sphere passing through (A), (B), (C), and (D). Since (O) is the circumcenter, it must satisfy (OA = OB = OC = OD = R).Given the symmetry, (O) must lie along the z-axis, so let me denote (O = (0, 0, k)) for some (k). Now, compute the distances from (O) to each vertex:- (OA = sqrt{(1 - 0)^2 + (0 - 0)^2 + (0 - k)^2} = sqrt{1 + k^2})- (OB = sqrt{(-frac{1}{2} - 0)^2 + (frac{sqrt{3}}{2} - 0)^2 + (0 - k)^2} = sqrt{frac{1}{4} + frac{3}{4} + k^2} = sqrt{1 + k^2})- (OC = sqrt{(-frac{1}{2} - 0)^2 + (-frac{sqrt{3}}{2} - 0)^2 + (0 - k)^2} = sqrt{frac{1}{4} + frac{3}{4} + k^2} = sqrt{1 + k^2})- (OD = sqrt{(0 - 0)^2 + (0 - 0)^2 + (h - k)^2} = |h - k|)Since (OA = OD), we have:(sqrt{1 + k^2} = |h - k|)Let me square both sides to eliminate the square root:(1 + k^2 = (h - k)^2)Expanding the right side:(1 + k^2 = h^2 - 2hk + k^2)Subtract (k^2) from both sides:(1 = h^2 - 2hk)Rearranged:(2hk = h^2 - 1)So,(k = frac{h^2 - 1}{2h})Now, let me note that (k) must be such that (O) is the circumcenter. Also, since (O) lies on the z-axis, this should be the only condition.Now, I need to find (cos angle AOB) and (cos angle AOD).First, let me find (cos angle AOB). The angle between vectors (OA) and (OB).Since (OA) and (OB) are both vectors from (O) to (A) and (B), respectively.Given (O = (0, 0, k)), (A = (1, 0, 0)), and (B = (-frac{1}{2}, frac{sqrt{3}}{2}, 0)).The vectors (OA) and (OB) are:(overrightarrow{OA} = (1, 0, -k))(overrightarrow{OB} = (-frac{1}{2}, frac{sqrt{3}}{2}, -k))The cosine of the angle between them is given by the dot product divided by the product of their magnitudes.First, compute the dot product:(overrightarrow{OA} cdot overrightarrow{OB} = (1)(-frac{1}{2}) + (0)(frac{sqrt{3}}{2}) + (-k)(-k) = -frac{1}{2} + k^2)The magnitudes of both vectors are equal since (OA = OB = R = sqrt{1 + k^2}).So,(cos angle AOB = frac{-frac{1}{2} + k^2}{( sqrt{1 + k^2} )^2} = frac{k^2 - frac{1}{2}}{1 + k^2})Simplify:(cos angle AOB = frac{k^2 - frac{1}{2}}{1 + k^2} = 1 - frac{frac{3}{2}}{1 + k^2})Wait, let me check that algebra:(frac{k^2 - frac{1}{2}}{1 + k^2} = frac{(1 + k^2) - frac{3}{2}}{1 + k^2} = 1 - frac{3}{2(1 + k^2)})Yes, that's correct.Now, let me find (cos angle AOD). The angle between vectors (OA) and (OD).Point (D = (0, 0, h)), so vector (OD) is ((0, 0, h - k)).Vector (OA) is ((1, 0, -k)).Compute the dot product:(overrightarrow{OA} cdot overrightarrow{OD} = (1)(0) + (0)(0) + (-k)(h - k) = -k(h - k))The magnitudes:(|overrightarrow{OA}| = sqrt{1 + k^2})(|overrightarrow{OD}| = |h - k|)So,(cos angle AOD = frac{-k(h - k)}{sqrt{1 + k^2} cdot |h - k|})Simplify:Since (h - k) is positive or negative depending on whether (h > k) or not, but since (O) is the circumcenter, it should lie inside the tetrahedron, so (k) is between 0 and (h). So, (h - k > 0).Thus,(cos angle AOD = frac{-k(h - k)}{sqrt{1 + k^2} cdot (h - k)} = frac{-k}{sqrt{1 + k^2}})So, (cos angle AOD = -frac{k}{sqrt{1 + k^2}})Now, let me write down both cosines:(cos angle AOB = 1 - frac{3}{2(1 + k^2)})(cos angle AOD = -frac{k}{sqrt{1 + k^2}})I need to find the sum:(cos angle AOB + cos angle AOD = 1 - frac{3}{2(1 + k^2)} - frac{k}{sqrt{1 + k^2}})Let me denote (t = sqrt{1 + k^2}). Then, (t geq 1), since (k) is real.Express everything in terms of (t):First, (k = sqrt{t^2 - 1})So,(cos angle AOB = 1 - frac{3}{2t^2})(cos angle AOD = -frac{sqrt{t^2 - 1}}{t})Thus, the sum becomes:(1 - frac{3}{2t^2} - frac{sqrt{t^2 - 1}}{t})Let me denote (f(t) = 1 - frac{3}{2t^2} - frac{sqrt{t^2 - 1}}{t}), where (t geq 1).I need to find the minimum value of (f(t)) for (t geq 1).To find the minimum, I can take the derivative of (f(t)) with respect to (t) and set it to zero.First, compute (f'(t)):(f'(t) = 0 - frac{d}{dt}left(frac{3}{2t^2}right) - frac{d}{dt}left(frac{sqrt{t^2 - 1}}{t}right))Compute each derivative:1. (frac{d}{dt}left(frac{3}{2t^2}right) = frac{3}{2} cdot (-2)t^{-3} = -frac{3}{t^3})2. (frac{d}{dt}left(frac{sqrt{t^2 - 1}}{t}right)):Let me denote (g(t) = frac{sqrt{t^2 - 1}}{t}). Then,(g(t) = frac{(t^2 - 1)^{1/2}}{t})Compute (g'(t)):Using the quotient rule:(g'(t) = frac{ frac{1}{2}(t^2 - 1)^{-1/2} cdot 2t cdot t - (t^2 - 1)^{1/2} cdot 1 }{t^2})Simplify numerator:( frac{t^2}{sqrt{t^2 - 1}} - sqrt{t^2 - 1} = frac{t^2 - (t^2 - 1)}{sqrt{t^2 - 1}} = frac{1}{sqrt{t^2 - 1}} )Thus,(g'(t) = frac{1}{sqrt{t^2 - 1} cdot t^2})So, putting it all together,(f'(t) = -(-frac{3}{t^3}) - frac{1}{sqrt{t^2 - 1} cdot t^2} = frac{3}{t^3} - frac{1}{t^2 sqrt{t^2 - 1}})Set (f'(t) = 0):(frac{3}{t^3} - frac{1}{t^2 sqrt{t^2 - 1}} = 0)Multiply both sides by (t^3 sqrt{t^2 - 1}):(3 sqrt{t^2 - 1} - t = 0)So,(3 sqrt{t^2 - 1} = t)Square both sides:(9(t^2 - 1) = t^2)(9t^2 - 9 = t^2)(8t^2 = 9)(t^2 = frac{9}{8})(t = frac{3}{2sqrt{2}} = frac{3sqrt{2}}{4})But wait, (t = sqrt{1 + k^2} geq 1), and (frac{3sqrt{2}}{4} approx 1.06), which is greater than 1, so it's valid.Now, let me check if this is a minimum.Compute the second derivative or test intervals.Alternatively, since it's the only critical point for (t > 1), and as (t to 1^+), (f(t)) approaches:(1 - frac{3}{2(1)} - frac{0}{1} = 1 - frac{3}{2} = -frac{1}{2})As (t to infty), (f(t)) approaches:(1 - 0 - 0 = 1)So, the function decreases from (t=1) to some point and then increases. Wait, but at (t = frac{3sqrt{2}}{4}), which is approximately 1.06, the derivative is zero. Let me check the behavior around this point.Wait, actually, when (t) increases beyond 1, initially, the function might be decreasing or increasing.Wait, let me compute (f'(t)) for (t) slightly greater than 1.Let me take (t = 1.1):Compute (f'(1.1)):(frac{3}{(1.1)^3} - frac{1}{(1.1)^2 sqrt{(1.1)^2 - 1}})Compute each term:1. (frac{3}{1.331} approx 2.255)2. (frac{1}{(1.21) sqrt{0.21}} approx frac{1}{1.21 times 0.458} approx frac{1}{0.554} approx 1.805)So, (f'(1.1) approx 2.255 - 1.805 = 0.45 > 0)So, at (t=1.1), the derivative is positive, meaning function is increasing.At (t = 2):Compute (f'(2)):(frac{3}{8} - frac{1}{4 sqrt{3}} approx 0.375 - 0.144 = 0.231 > 0)Still positive.Wait, but we found a critical point at (t = frac{3sqrt{2}}{4} approx 1.06), but when (t=1.1), derivative is positive, meaning function is increasing. So, perhaps the critical point is a minimum?Wait, let me compute (f'(t)) just below (t = frac{3sqrt{2}}{4}), say (t=1.05):Compute (f'(1.05)):(frac{3}{(1.05)^3} - frac{1}{(1.05)^2 sqrt{(1.05)^2 - 1}})Compute each term:1. (frac{3}{1.157625} approx 2.591)2. (frac{1}{(1.1025) sqrt{0.1025}} approx frac{1}{1.1025 times 0.32} approx frac{1}{0.3528} approx 2.835)So, (f'(1.05) approx 2.591 - 2.835 = -0.244 < 0)So, for (t < frac{3sqrt{2}}{4}), derivative is negative, and for (t > frac{3sqrt{2}}{4}), derivative is positive. Therefore, the function has a minimum at (t = frac{3sqrt{2}}{4}).Thus, the minimum value of (f(t)) is at (t = frac{3sqrt{2}}{4}).Compute (fleft(frac{3sqrt{2}}{4}right)):First, compute (t = frac{3sqrt{2}}{4}), so (t^2 = frac{9 times 2}{16} = frac{18}{16} = frac{9}{8}).Compute (f(t)):(1 - frac{3}{2t^2} - frac{sqrt{t^2 - 1}}{t})Substitute (t^2 = frac{9}{8}):(1 - frac{3}{2 times frac{9}{8}} - frac{sqrt{frac{9}{8} - 1}}{frac{3sqrt{2}}{4}})Simplify each term:1. (1)2. (frac{3}{2 times frac{9}{8}} = frac{3}{frac{9}{4}} = frac{3 times 4}{9} = frac{12}{9} = frac{4}{3})3. (sqrt{frac{9}{8} - 1} = sqrt{frac{1}{8}} = frac{sqrt{2}}{4})Thus, the third term:(frac{frac{sqrt{2}}{4}}{frac{3sqrt{2}}{4}} = frac{sqrt{2}}{4} times frac{4}{3sqrt{2}} = frac{1}{3})Putting it all together:(1 - frac{4}{3} - frac{1}{3} = 1 - frac{5}{3} = -frac{2}{3})So, the minimum value of (f(t)) is (-frac{2}{3}), which occurs at (t = frac{3sqrt{2}}{4}).Therefore, (cos angle AOB + cos angle AOD geq -frac{2}{3}), as required.**Final Answer**boxed{-dfrac{2}{3}}

💡To translate the equation ( y = 2x - 1 ) upward by 2 units, we add 2 to the entire equation. This results in the new equation ( y = 2x - 1 + 2 ), which simplifies to ( y = 2x + 1 ).

💡Okay, so I have this equation to solve in integers: (5x^2 + y^2 + 3z^2 - 2yz = 30). Hmm, that looks a bit complicated, but maybe I can break it down. Let me see... First, I notice that there are three variables: (x), (y), and (z). Since it's a quadratic equation, maybe I can find some way to simplify it or factor it. Let me try to rearrange the terms to see if something stands out.Looking at the equation: (5x^2 + y^2 + 3z^2 - 2yz = 30). I see that there are terms with (y) and (z) that might be combinable. Specifically, (y^2 - 2yz + z^2) looks like a perfect square. Let me check that: (y^2 - 2yz + z^2 = (y - z)^2). Oh, that's right! So I can rewrite the equation by completing the square for (y) and (z).So, substituting that in, the equation becomes:[5x^2 + (y - z)^2 + 2z^2 = 30]Okay, that seems a bit simpler. Now, I have (5x^2) plus a square term plus another term with (z^2). Maybe I can find bounds on (x) first because (x) is multiplied by 5, which might limit its possible values.Let's think about the possible values of (x). Since (5x^2) is part of the equation and the total is 30, (5x^2) can't be more than 30. So:[5x^2 leq 30]Dividing both sides by 5:[x^2 leq 6]Taking the square root:[|x| leq sqrt{6}]Which is approximately 2.45. Since (x) has to be an integer, the possible values for (x) are (-2, -1, 0, 1, 2).Alright, so I can consider each possible value of (x) and see if I can find corresponding integers (y) and (z) that satisfy the equation.Let me start with (x = 0):Substituting (x = 0) into the equation:[(y - z)^2 + 2z^2 = 30]Now, I need to find integers (y) and (z) such that this holds. Let's see what possible values (z) can take. Since (2z^2) is part of the equation, (2z^2 leq 30), so:[z^2 leq 15]Which means (|z| leq sqrt{15}), approximately 3.87. So (z) can be (-3, -2, -1, 0, 1, 2, 3).Let me check each possible (z):1. (z = 0): [(y - 0)^2 + 0 = 30] So (y^2 = 30). But 30 isn't a perfect square, so no integer (y) here.2. (z = pm 1): [(y mp 1)^2 + 2 = 30] So ((y mp 1)^2 = 28). 28 isn't a perfect square either, so no solution.3. (z = pm 2): [(y mp 2)^2 + 8 = 30] So ((y mp 2)^2 = 22). 22 isn't a perfect square, no solution.4. (z = pm 3): [(y mp 3)^2 + 18 = 30] So ((y mp 3)^2 = 12). 12 isn't a perfect square, so no solution.Hmm, so for (x = 0), there are no integer solutions for (y) and (z). Let's move on to (x = 1).Substituting (x = 1):[5(1)^2 + (y - z)^2 + 2z^2 = 30]Which simplifies to:[5 + (y - z)^2 + 2z^2 = 30]Subtracting 5 from both sides:[(y - z)^2 + 2z^2 = 25]Again, let's find possible (z) values. (2z^2 leq 25), so:[z^2 leq 12.5]Which means (|z| leq 3) (since (3^2 = 9) and (4^2 = 16) which is too big).So (z) can be (-3, -2, -1, 0, 1, 2, 3). Let's check each:1. (z = 0): [(y - 0)^2 + 0 = 25] So (y^2 = 25), which gives (y = pm 5). So we have solutions ((1, 5, 0)) and ((1, -5, 0)).2. (z = pm 1): [(y mp 1)^2 + 2 = 25] So ((y mp 1)^2 = 23). 23 isn't a perfect square, so no solution.3. (z = pm 2): [(y mp 2)^2 + 8 = 25] So ((y mp 2)^2 = 17). 17 isn't a perfect square, no solution.4. (z = pm 3): [(y mp 3)^2 + 18 = 25] So ((y mp 3)^2 = 7). 7 isn't a perfect square, so no solution.Alright, so for (x = 1), we have two solutions: ((1, 5, 0)) and ((1, -5, 0)).Now, let's check (x = 2):Substituting (x = 2):[5(2)^2 + (y - z)^2 + 2z^2 = 30]Which simplifies to:[20 + (y - z)^2 + 2z^2 = 30]Subtracting 20:[(y - z)^2 + 2z^2 = 10]Again, find possible (z) values. (2z^2 leq 10), so:[z^2 leq 5]Which means (|z| leq 2) (since (2^2 = 4) and (3^2 = 9) is too big).So (z) can be (-2, -1, 0, 1, 2). Let's check each:1. (z = 0): [(y - 0)^2 + 0 = 10] So (y^2 = 10). Not a perfect square, no solution.2. (z = pm 1): [(y mp 1)^2 + 2 = 10] So ((y mp 1)^2 = 8). 8 isn't a perfect square, no solution.3. (z = pm 2): [(y mp 2)^2 + 8 = 10] So ((y mp 2)^2 = 2). 2 isn't a perfect square, no solution.So, no solutions for (x = 2).Now, what about negative values of (x)? Since (x) is squared, (x = -1) and (x = -2) will give the same results as (x = 1) and (x = 2). Let me check (x = -1):Substituting (x = -1):[5(-1)^2 + (y - z)^2 + 2z^2 = 30]Which is the same as:[5 + (y - z)^2 + 2z^2 = 30]So, same as (x = 1), leading to solutions ((-1, 5, 0)) and ((-1, -5, 0)).Similarly, (x = -2) would lead to the same equation as (x = 2), which had no solutions.So, putting it all together, the integer solutions are:[(1, 5, 0), (1, -5, 0), (-1, 5, 0), (-1, -5, 0)]I think that's all the possible solutions because we've checked all possible (x) values and corresponding (z) values, and only these combinations worked out.

💡First, I need to determine the total number of active volcanoes in the mountain range. Let's denote this number as ( V ).In the first two months, 20% of the volcanoes exploded. This means that 80% of the volcanoes remained intact. I can express this as:[0.8V]By the halfway point of the year, 40% of the remaining volcanoes exploded. Therefore, 60% of the remaining volcanoes were still intact. This can be represented as:[0.6 times 0.8V = 0.48V]At the end of the year, another 50% of the remaining volcanoes exploded, leaving 50% of the remaining volcanoes intact. This is calculated as:[0.5 times 0.48V = 0.24V]We know that there are 48 mountains still intact at the end of the year. This means:[0.24V = 48]To find ( V ), I divide both sides of the equation by 0.24:[V = frac{48}{0.24} = 200]Therefore, there are 200 active volcanoes in the mountain range.

💡First, I need to determine the total surface area of the cube. A cube has 6 faces, and each face is a square with sides of 10 feet. Calculating the area of one face:10 feet × 10 feet = 100 square feet.Since there are 6 faces:6 × 100 square feet = 600 square feet.Next, I know that one quart of paint covers 1200 square feet. To find out how much paint is needed for 600 square feet:600 ÷ 1200 = 0.5 quarts.The cost to paint the cube is 1.60 for 0.5 quarts. To find the cost per quart:1.60 ÷ 0.5 = 3.20 per quart.