answer:1. **Let Variables Represent the Problem:** - Let ( A, B, C, ) and ( D ) represent Anya, Vanya, Danya, and Tanya respectively. - Let ( x_A, x_B, x_C, ) and ( x_D ) be the number of apples collected by each of them. - Let ( n ) be the total number of apples collected. 2. **Define the Conditions:** - Each person collected a distinct integer percentage of the total number of apples. - These percentages add up to 100%, so we have ( a% + b% + c% + d% = 100% ). - Rearrange the variables if necessary such that ( a < b < c < d ). 3. **Smallest Total Collection:** - The problem statement suggests that Tanya collected more apples than anyone else. - Since ( d % ) must be the largest single percentage and still need to agree with distinct integer values summing to 100%, plausible values that add up and satisfy other constraints should be considered. 4. **Initial Assumption and Setup:** - Let ( k ) be the percentage of the smallest collector. - We have four distinct integer percentages ( k, m, p, q ) such that ( k < m < p < q ) and ( k + m + p + q = 100 ). - Minimizing ( k ) to find the smallest possible total number of apples ( n ). 5. **Minimizing the Collection:** - If we assume ( k = 10% ), ( m, p, ) and ( q ) must be distinct integers that sum with ( k ) to 100. - Typical integer sets working under this rule are explored. Let's try ( k = 10, m = 15, p = 20, ) and ( q = 55 ), effectively yielding distinct values summing to 100%. 6. **Total Apples Distribution:** - Let us confirm this in a minimum distribution, ( n = 20 ) (total minimum feasible for clear division by percent: - ( k = 10 %: frac{10}{100} times 20 = 2 ) - ( m = 15 %: frac{15}{100} times 20 = 3 ) - ( p = 20 %: frac{20}{100} times 20 = 4 ) - ( q = 55 %: frac{55}{100} times 20 = 11 ) - This assigns: - Anya: 2 apples ( (approx 10%) ), - Vanya: 3 apples ( (approx 15%) ), - Danya: 4 apples ( (approx 20%) ), - Tanya: 11 apples ( (approx 55%) ). 7. **Verification and Outcome:** - After Tanya eats her apples, the remaining apples are ( 20 - 11 = 9 ). - Remaining percent should align with others: - Anya ( frac{2}{9}, approx 22.22% ), - Vanya ( frac{3}{9}, approx 33.33% ), - Danya ( frac{4}{9}, approx 44.44%). Since integer values must remain consistent in initial context and set tenth inputs, the minimal ( boxed{20} ).

answer:To ensure the function is well-defined, each expression under a square root must be non-negative. We start with the innermost square root: 1. For sqrt{x} to be real and non-negative, we require: x geq 0. 2. For sqrt{6 - sqrt{x}} to be real and non-negative, we need: 6 - sqrt{x} geq 0 sqrt{x} leq 6 x leq 36. 3. For sqrt{4 - sqrt{6 - sqrt{x}}} to be real and non-negative, we require: 4 - sqrt{6 - sqrt{x}} geq 0 sqrt{6 - sqrt{x}} leq 4. Squaring both sides and solving for x, we get: 6 - sqrt{x} leq 16 sqrt{x} geq -10, which is always true since sqrt{x} is non-negative. Thus, we consider only: sqrt{x} geq -10 which implies no further restrictions on x beyond those already stated. Combining all these, the domain of x is from the intersection of x geq 0 and x leq 36. Hence, the domain of f(x) is: boxed{[0, 36]}

answer:To calculate the total area of the garden, we need to find the area of the trapezoids and the triangles and then sum them up. The area of a trapezoid can be calculated using the formula: [ text{Area of trapezoid} = frac{1}{2} times (b_1 + b_2) times h ] where ( b_1 ) and ( b_2 ) are the lengths of the two bases, and ( h ) is the height. For the trapezoids in the garden: [ b_1 = 15 text{ meters} ] [ b_2 = 12 text{ meters} ] [ h = 3 text{ meters} ] Plugging these values into the formula, we get: [ text{Area of one trapezoid} = frac{1}{2} times (15 + 12) times 3 ] [ text{Area of one trapezoid} = frac{1}{2} times 27 times 3 ] [ text{Area of one trapezoid} = frac{1}{2} times 81 ] [ text{Area of one trapezoid} = 40.5 text{ square meters} ] Since there are 4 trapezoids, the total area of the trapezoids is: [ 4 times 40.5 = 162 text{ square meters} ] The area of a triangle can be calculated using the formula: [ text{Area of triangle} = frac{1}{2} times b times h ] where ( b ) is the length of the base, and ( h ) is the height. For the triangles in the garden: [ b = 7 text{ meters} ] [ h = 4 text{ meters} ] Plugging these values into the formula, we get: [ text{Area of one triangle} = frac{1}{2} times 7 times 4 ] [ text{Area of one triangle} = frac{1}{2} times 28 ] [ text{Area of one triangle} = 14 text{ square meters} ] Since there are 4 triangles, the total area of the triangles is: [ 4 times 14 = 56 text{ square meters} ] Now, we can sum up the total area of the trapezoids and the triangles to get the total area of the garden: [ text{Total area of the garden} = text{Area of trapezoids} + text{Area of triangles} ] [ text{Total area of the garden} = 162 + 56 ] [ text{Total area of the garden} = 218 text{ square meters} ] Therefore, the total area of the garden is boxed{218} square meters.

answer:To solve the given system of inequalities for x, we proceed as follows: 1. For the first inequality, x + 5 > 0, we solve for x: [x > -5] 2. For the second inequality, x - m leq 1, we solve for x: [x leq m + 1] Given that the system has 3 integer solutions, we need to identify these solutions within the constraints provided by the inequalities. 3. The integers greater than -5 and closest to it are -4, -3, and -2. These are the 3 integer solutions that satisfy the first inequality. 4. To ensure these solutions also satisfy the second inequality (x leq m + 1), we substitute x = -2, the largest of the three solutions, into the inequality: [-2 leq m + 1] Solving for m gives: [m geq -3] 5. However, to maintain exactly 3 integer solutions, the next integer solution, -1, must not satisfy the inequality. Therefore, we also consider: [-1 notleq m + 1] Solving for m gives: [m < -2] Combining these conditions, we find the range of m that allows exactly 3 integer solutions to the system: [-3 leq m < -2] Therefore, the range of the real number m for which the inequality system has 3 integer solutions is boxed{-3 leq m < -2}.