💡First, calculate the number of strings needed for the basses. John has to restring 5 basses, each with 4 strings:5 basses × 4 strings/bass = 20 strings.Next, determine the number of normal guitars. There are three times as many guitars as basses:3 × 5 basses = 15 guitars.Calculate the number of strings for the normal guitars. Each guitar has 6 strings:15 guitars × 6 strings/guitar = 90 strings.Find out the number of 8-string guitars, which is 2 fewer than the normal guitars:15 guitars - 2 = 13 8-string guitars.Calculate the number of strings for the 8-string guitars:13 8-string guitars × 8 strings/guitar = 104 strings.Determine the number of 12-string guitars, which is the same as the number of basses:5 12-string guitars.Calculate the number of strings for the 12-string guitars:5 12-string guitars × 12 strings/guitar = 60 strings.Finally, add up all the strings needed:20 strings (basses) + 90 strings (guitars) + 104 strings (8-string guitars) + 60 strings (12-string guitars) = 274 strings.

💡Okay, so I have this problem where I need to plot the set of points (p, q) such that the quadratic equation x² + 2px + q = 0 has two roots, one greater than 2 and the other less than 0. Hmm, quadratic equations and their roots... I remember that for a quadratic equation ax² + bx + c = 0, the roots can be found using the quadratic formula, right? So, maybe I should start by recalling that formula.The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a). In this case, the equation is x² + 2px + q = 0, so a = 1, b = 2p, and c = q. Plugging these into the quadratic formula, the roots would be x = [-2p ± sqrt((2p)² - 4*1*q)] / 2. Simplifying that, it becomes x = [-2p ± sqrt(4p² - 4q)] / 2, which further simplifies to x = -p ± sqrt(p² - q). Okay, so the two roots are -p + sqrt(p² - q) and -p - sqrt(p² - q).Now, the problem states that one root is greater than 2 and the other is less than 0. So, I need to set up inequalities for these roots. Let me denote the roots as r1 and r2, where r1 = -p + sqrt(p² - q) and r2 = -p - sqrt(p² - q). Without loss of generality, let's assume r1 > 2 and r2 < 0.So, first inequality: r1 > 2. That is, -p + sqrt(p² - q) > 2. Let me write that down: -p + sqrt(p² - q) > 2. Second inequality: r2 < 0, which is -p - sqrt(p² - q) < 0.I need to solve these inequalities for p and q. Let me start with the first inequality: -p + sqrt(p² - q) > 2. Let me rearrange this: sqrt(p² - q) > p + 2. Hmm, sqrt(p² - q) is a square root, so it's always non-negative. Therefore, the right side p + 2 must also be positive for this inequality to hold, otherwise, the inequality would automatically be true if p + 2 is negative. Wait, so if p + 2 is negative, then sqrt(p² - q) > p + 2 is always true because sqrt(p² - q) is non-negative and p + 2 is negative. So, in that case, the inequality doesn't give us any additional information. But if p + 2 is positive, then we can square both sides to get rid of the square root.So, let's consider two cases:Case 1: p + 2 ≤ 0, which implies p ≤ -2. In this case, sqrt(p² - q) > p + 2 is always true because the left side is non-negative and the right side is non-positive. So, in this case, the first inequality doesn't impose any additional constraints.Case 2: p + 2 > 0, which implies p > -2. In this case, we can square both sides: (sqrt(p² - q))² > (p + 2)², which simplifies to p² - q > p² + 4p + 4. Subtracting p² from both sides, we get -q > 4p + 4, which can be rewritten as q < -4p - 4.Okay, so from the first inequality, we have that if p ≤ -2, then there's no additional constraint, but if p > -2, then q must be less than -4p - 4.Now, let's look at the second inequality: -p - sqrt(p² - q) < 0. Let's rearrange this: -p < sqrt(p² - q). Again, sqrt(p² - q) is non-negative, so the right side is non-negative. The left side is -p, so depending on the value of p, this can be positive or negative.Let me consider two cases again:Case 1: -p < 0, which implies p > 0. In this case, the inequality -p < sqrt(p² - q) is automatically true because the left side is negative and the right side is non-negative.Case 2: -p ≥ 0, which implies p ≤ 0. In this case, both sides are non-negative, so we can square both sides: (-p)² < (sqrt(p² - q))², which simplifies to p² < p² - q. Subtracting p² from both sides, we get 0 < -q, which implies q < 0.So, from the second inequality, if p > 0, there's no additional constraint, but if p ≤ 0, then q must be less than 0.Now, let's combine the results from both inequalities.From the first inequality:- If p ≤ -2, no additional constraint from the first inequality.- If p > -2, then q < -4p - 4.From the second inequality:- If p > 0, no additional constraint from the second inequality.- If p ≤ 0, then q < 0.So, let's consider the regions:1. When p ≤ -2: - From the first inequality: no constraint. - From the second inequality: since p ≤ -2 ≤ 0, we have q < 0. - So, in this region, q < 0.2. When -2 < p ≤ 0: - From the first inequality: q < -4p - 4. - From the second inequality: q < 0. - So, in this region, both conditions must be satisfied. That is, q must be less than the minimum of (-4p - 4) and 0. Since -4p - 4 when p = -2 is 0, and as p increases from -2 to 0, -4p - 4 decreases from 0 to -4. So, in this interval, -4p - 4 is less than 0. Therefore, q < -4p - 4.3. When p > 0: - From the first inequality: q < -4p - 4. But since p > 0, -4p - 4 is less than -4, which is negative. - From the second inequality: no constraint. - So, in this region, q < -4p - 4.Wait a minute, but when p > 0, q < -4p - 4, which is a line with a negative slope. However, we also need to ensure that the quadratic equation has real roots, meaning the discriminant must be non-negative. The discriminant is (2p)² - 4*1*q = 4p² - 4q ≥ 0, which simplifies to p² - q ≥ 0, or q ≤ p².So, in addition to the previous conditions, we must have q ≤ p².Therefore, combining all these:- For p ≤ -2: q < 0 and q ≤ p². Since p² is always non-negative, and q < 0, the condition q < 0 is more restrictive.- For -2 < p ≤ 0: q < -4p - 4 and q ≤ p². We need to check where -4p - 4 is less than p².Let me solve -4p - 4 ≤ p² for p in (-2, 0].Rearranging: p² + 4p + 4 ≥ 0. This factors to (p + 2)² ≥ 0, which is always true. So, in this interval, -4p - 4 ≤ p², so the condition q < -4p - 4 is more restrictive.- For p > 0: q < -4p - 4 and q ≤ p². Again, we need to check where -4p - 4 ≤ p².Solving -4p - 4 ≤ p²: p² + 4p + 4 ≥ 0, which is (p + 2)² ≥ 0, always true. So, q < -4p - 4 is more restrictive.However, for p > 0, q < -4p - 4 is negative, but q ≤ p² is positive. So, in this case, q must satisfy both, but since q < -4p - 4 is negative and q ≤ p² is positive, the intersection is q < -4p - 4.Wait, but if p > 0, q < -4p - 4 is negative, but q ≤ p² is positive. So, actually, for p > 0, q must be less than -4p - 4, which is negative, but also q ≤ p², which is positive. So, the intersection is q < -4p - 4, since that's more restrictive.But hold on, if p > 0, q < -4p - 4 is negative, but q ≤ p² is positive. So, actually, for p > 0, q must be less than -4p - 4, which is negative, but also q ≤ p², which is positive. So, the intersection is q < -4p - 4, since that's more restrictive.But wait, if p > 0, q < -4p - 4 is negative, but q ≤ p² is positive. So, actually, for p > 0, q must be less than -4p - 4, which is negative, but also q ≤ p², which is positive. So, the intersection is q < -4p - 4, since that's more restrictive.But hold on, if p > 0, q < -4p - 4 is negative, but q ≤ p² is positive. So, actually, for p > 0, q must be less than -4p - 4, which is negative, but also q ≤ p², which is positive. So, the intersection is q < -4p - 4, since that's more restrictive.Wait, I think I'm repeating myself. Let me try to summarize:The region we're looking for is the set of (p, q) such that:- If p ≤ -2: q < 0.- If -2 < p ≤ 0: q < -4p - 4.- If p > 0: q < -4p - 4.Additionally, we must have q ≤ p² to ensure the quadratic has real roots.So, plotting this on the coordinate plane:1. For p ≤ -2: q < 0. This is the region below the p-axis (q=0) for p ≤ -2.2. For -2 < p ≤ 0: q < -4p - 4. This is a line with slope -4 and y-intercept -4. For p between -2 and 0, this line goes from (p=-2, q=0) to (p=0, q=-4). The region below this line.3. For p > 0: q < -4p - 4. This continues the line beyond p=0, going into negative q values as p increases.Additionally, we must ensure q ≤ p². So, the region is also below the parabola q = p².Therefore, the set of points (p, q) is the intersection of the regions defined above and below the parabola q = p².So, to plot this, I would:- Draw the p-q coordinate plane.- Plot the line q = -4p - 4. It starts at (p=-2, q=0) and goes downward as p increases.- Shade the region below this line for p > -2.- For p ≤ -2, shade the region below q=0.- Also, plot the parabola q = p² and shade the region below it.- The intersection of these shaded regions is the desired set.I think that's the general idea. Let me double-check if I missed anything.Wait, when p ≤ -2, q < 0, but also q ≤ p². Since p² is positive, and q < 0, the condition q < 0 is more restrictive. So, for p ≤ -2, the region is q < 0.For -2 < p ≤ 0, q < -4p - 4, which is below the line, and also q ≤ p². Since -4p - 4 ≤ p² in this interval, the region is below the line.For p > 0, q < -4p - 4, which is below the line, and also q ≤ p². Since -4p - 4 is negative and p² is positive, the region is below the line.So, yes, the final region is the union of these areas.I think that's it. So, the set of points (p, q) is the region below the line q = -4p - 4 for p > -2 and below q=0 for p ≤ -2, all while being below the parabola q = p².

💡First, I need to calculate the time each runner takes for their respective 200-meter sections, considering the terrain, obstacles, and their handicaps.For Rhonda:- Her usual time for 200 meters is 24 seconds.- The steep incline adds 2 seconds to her time.- Her total time before applying the handicap is 24 + 2 = 26 seconds.- Applying her handicap factor of 0.95: 26 * 0.95 = 24.7 seconds.For Sally:- Her usual time for 200 meters is 26 seconds.- The hurdles add 1 second per hurdle, and there are five hurdles, adding 5 seconds.- Her total time before applying the handicap is 26 + 5 = 31 seconds.- Applying her handicap factor of 0.90: 31 * 0.90 = 27.9 seconds.For Diane:- Her usual time for 200 meters is 21 seconds.- The muddy terrain slows her down by 10%, adding 2.1 seconds.- Her total time before applying the handicap is 21 + 2.1 = 23.1 seconds.- Applying her handicap factor of 1.05: 23.1 * 1.05 = 24.255 seconds.Finally, I add up the final times of all three runners to get the total time for the 600-meter relay race:24.7 + 27.9 + 24.255 = 76.855 seconds.

💡First, I need to determine how much each person earned before taxes.Salvadore earned 1956.Santo earned half of Salvadore's earnings:Santo's earnings = 1/2 * 1956 = 978.Maria earned three times Santo's earnings:Maria's earnings = 3 * 978 = 2934.Pedro earned the combined amount of Santo and Maria:Pedro's earnings = 978 + 2934 = 3912.Next, I'll calculate the taxes each person paid and their earnings after taxes.Salvadore paid 20% in taxes:Salvadore's taxes = 0.20 * 1956 = 391.20.Salvadore's earnings after taxes = 1956 - 391.20 = 1564.80.Santo paid 15% in taxes:Santo's taxes = 0.15 * 978 = 146.70.Santo's earnings after taxes = 978 - 146.70 = 831.30.Maria paid 10% in taxes:Maria's taxes = 0.10 * 2934 = 293.40.Maria's earnings after taxes = 2934 - 293.40 = 2640.60.Pedro paid 25% in taxes:Pedro's taxes = 0.25 * 3912 = 978.Pedro's earnings after taxes = 3912 - 978 = 2934.Finally, I'll sum up their earnings after taxes to find the total amount earned together.Total earnings after taxes = 1564.80 + 831.30 + 2640.60 + 2934 = 7971.70.

💡Okay, so I have this function g(x) which is equal to (x + 5) divided by (x squared plus c times x plus d). I need to find the constants c and d such that the function has vertical asymptotes at x equals -1 and x equals 3. Then, I have to compute the sum of c and d. Hmm, vertical asymptotes... I remember that vertical asymptotes occur where the denominator is zero, provided that the numerator isn't also zero at those points. So, if there are vertical asymptotes at x = -1 and x = 3, that means the denominator must be zero when x is -1 or 3. Alright, so the denominator is a quadratic equation: x² + c x + d. If the denominator is zero at x = -1 and x = 3, that means these are the roots of the quadratic equation. So, I can write the denominator in its factored form as (x + 1)(x - 3) because if x is -1, then (x + 1) becomes zero, and if x is 3, then (x - 3) becomes zero. Let me expand this factored form to get it into standard quadratic form. So, (x + 1)(x - 3) equals x times x, which is x², then x times -3 is -3x, then 1 times x is x, and 1 times -3 is -3. So, combining like terms, that's x² - 3x + x - 3. The -3x and +x combine to -2x, so the expanded form is x² - 2x - 3. Therefore, the denominator x² + c x + d must be equal to x² - 2x - 3. So, comparing the coefficients, the coefficient of x² is 1 in both, so that's consistent. The coefficient of x in the denominator is c, and in the expanded form, it's -2. So, c must be -2. Similarly, the constant term d is -3 because in the expanded form, the constant term is -3. So, c is -2 and d is -3. The problem asks for the sum of c and d. So, adding them together: -2 + (-3) equals -5. Wait, let me double-check to make sure I didn't make any mistakes. If I plug x = -1 into the denominator x² + c x + d, it should be zero. Plugging in x = -1: (-1)² + c*(-1) + d = 1 - c + d. If c is -2 and d is -3, then 1 - (-2) + (-3) is 1 + 2 - 3, which is 0. That works. Similarly, plugging in x = 3: 3² + c*3 + d = 9 + 3c + d. With c = -2 and d = -3, that's 9 + (-6) + (-3) = 0. That also works. So, my values for c and d seem correct.I think I've got it. The sum of c and d is -5.

💡Okay, so I need to find the simple interest on 10,000 at 4% per annum for 12 months. Hmm, I remember that simple interest is calculated using a formula, but I'm not entirely sure about the details. Let me think about what I know.First, I think the formula for simple interest is something like Principal multiplied by Rate multiplied by Time. Yeah, that sounds right. So, in symbols, it would be SI = P × R × T. Okay, so I need to identify the principal, the rate, and the time.The principal is the initial amount of money, which in this case is 10,000. That seems straightforward. The rate is given as 4% per annum. I know that percentages need to be converted to decimals for calculations, so 4% would be 0.04. Got that down.Now, the time is 12 months. But wait, the rate is per annum, which means it's annual. So, I need to make sure the time is in years. Since 12 months is exactly one year, that should be easy. So, T is 1 year.Putting it all together, the formula should look like SI = 10,000 × 0.04 × 1. Let me do the math step by step. First, multiply the principal by the rate: 10,000 × 0.04. That should give me 400. Then, multiply that by the time, which is 1. So, 400 × 1 is still 400.Wait, does that make sense? If I have 10,000 earning 4% interest over one year, I should get 400 in interest. That seems reasonable. I don't think I missed anything here. The time was already in years, so I didn't have to convert it. The rate was correctly converted to a decimal.I guess another way to check is to think about what 4% of 10,000 is. 1% of 10,000 is 100, so 4% would be 400. Yeah, that matches what I calculated earlier. So, I'm pretty confident that the simple interest is 400.Just to recap, the steps were:1. Identify the principal amount: 10,000.2. Convert the interest rate from percentage to decimal: 4% becomes 0.04.3. Determine the time in years: 12 months is 1 year.4. Plug these values into the simple interest formula: SI = P × R × T.5. Calculate the result: 10,000 × 0.04 × 1 = 400.Everything seems to add up, so I think I've got the right answer.