answer:Given the function fleft(xright)=3x^{3}+2ax^{2}+left(2+aright)x, its derivative is calculated as follows: 1. Differentiate f(x) to find f'(x): [f'(x) = frac{d}{dx}(3x^{3}) + frac{d}{dx}(2ax^{2}) + frac{d}{dx}((2+a)x)] [= 9x^{2} + 4ax + 2 + a] Since f'(x) is an even function, it must satisfy the property that f'(-x) = f'(x): 2. Apply the even function property: [f'(-x) = 9(-x)^{2} + 4a(-x) + 2 + a] [= 9x^{2} - 4ax + 2 + a] Setting f'(-x) = f'(x), we get: [9x^{2} - 4ax + 2 + a = 9x^{2} + 4ax + 2 + a] 3. Solve for a: [9x^{2} - 4ax + 2 + a = 9x^{2} + 4ax + 2 + a] Simplifying, we find that -4ax = 4ax, which implies a = 0. With a = 0, update f(x) and f'(x): [f(x) = 3x^{3} + 2(0)x^{2} + (2+0)x = 3x^{3} + 2x] [f'(x) = 9x^{2} + 2] 4. Calculate f(1) and f'(1) for the tangent line equation: [f(1) = 3(1)^{3} + 2(1) = 3 + 2 = 5] [f'(1) = 9(1)^{2} + 2 = 9 + 2 = 11] 5. Write the equation of the tangent line at left(1, f(1)right): The slope-intercept form is y - y_1 = m(x - x_1), where m = f'(1) and (x_1, y_1) = (1, 5): [y - 5 = 11(x - 1)] 6. Simplify the equation: [y - 5 = 11x - 11] [11x - y - 6 = 0] Therefore, the correct answer is encapsulated as boxed{A: 11x-y-6=0}.

answer:**Step 1: Factorize 9720 into its prime factors** To find the sum of the distinct prime factors of 9720, we begin by breaking it down into prime factors. We start with the smallest prime numbers. - **Divisibility by 2**: 9720 is even, divisible by 2: [ 9720 div 2 = 4860, quad 4860 div 2 = 2430, quad 2430 div 2 = 1215 ] So, 9720 = 2^3 times 1215. - **Divisibility by 3**: 1215 ends in 5, which isn't a criterion, but the sum of its digits (1+2+1+5=9) is divisible by 3: [ 1215 div 3 = 405, quad 405 div 3 = 135, quad 135 div 3 = 45, quad 45 div 3 = 15, quad 15 div 3 = 5 ] So, 1215 = 3^4 times 5. Combining these results, the complete prime factorization of 9720 is: [ 9720 = 2^3 times 3^4 times 5 ] **Step 2: Identify the distinct prime factors** The distinct prime factors of 9720 are 2, 3, and 5. **Step 3: Calculate the sum of the distinct prime factors** We add these distinct prime factors together: [ 2 + 3 + 5 = 10 ] **Conclusion with boxed answer:** The sum of the distinct prime integer divisors of 9720 is 10. The final answer is boxed{C) 10}

answer:To find Nicky's pace, we need to determine how far he ran in the 36 seconds before Cristina caught up to him. Since Cristina gave Nicky a 36-meter head start and caught up to him in 36 seconds, she must have run 36 meters more than Nicky in that time. Cristina runs at a pace of 4 meters per second, so in 36 seconds, she would have run: ( 4 text{ meters/second} times 36 text{ seconds} = 144 text{ meters} ) Since Cristina ran 144 meters to catch up to Nicky, and this distance includes the 36-meter head start, we can calculate the distance Nicky ran as follows: ( 144 text{ meters} - 36 text{ meters} = 108 text{ meters} ) Now we know Nicky ran 108 meters in 36 seconds. To find Nicky's pace in meters per second, we divide the distance he ran by the time it took: ( text{Nicky's pace} = frac{108 text{ meters}}{36 text{ seconds}} ) ( text{Nicky's pace} = 3 text{ meters/second} ) So, Nicky's pace is boxed{3} meters per second.

answer:Let b be the number of bagels and m be the number of muffins. Over the seven-day period, we have: [ b + m = 7 ] The cost of a bagel is 65 cents and the cost of a muffin is 40 cents. The total cost for the week, in cents, is: [ 65b + 40m ] The total cost must be divisible by 100. Using the equation m = 7 - b: [ 65b + 40(7 - b) = 65b + 280 - 40b = 25b + 280 ] We need 25b + 280 to be divisible by 100. Simplifying, we find: [ 25b + 280 equiv 0 pmod{100} ] [ 25b equiv -280 pmod{100} ] [ 25b equiv 20 pmod{100} ] [ b equiv 0.8 pmod{4} ] Considering integer values for b within the range 0 to 7, we solve for b. The equation 25b + 280 = 100k (where k is an integer) simplifies to: [ 25b = 100k - 280 ] [ b = frac{100k - 280}{25} ] Checking for integer solutions, we find k = 3 gives b = 2.8 (not an integer), k = 4 gives b = 4.8 (not an integer), continuing, we find: - For k = 3, b = 3 gives 25 cdot 3 + 280 = 355, not divisible by 100. - For k = 4, b = 4 gives 25 cdot 4 + 280 = 380, divisible by 100. When b = 4, the total number of muffins m = 3, and the total cost is 65 cdot 4 + 40 cdot 3 = 260 + 120 = 380 cents, or 3.8 dollars, hence b = 4 as the integers that makes the total cost a whole number of dollars. The final answer is boxed{C) 4}