💡First, I need to determine the number of face-masks Manolo makes during each specific time period of his eight-hour shift.1. **First Hour:** - Manolo makes one face-mask every 4 minutes. - There are 60 minutes in an hour. - Number of masks = 60 minutes / 4 minutes per mask = 15 masks.2. **Second and Third Hours:** - Manolo makes one face-mask every 6 minutes. - This period lasts for 2 hours, totaling 120 minutes. - Number of masks = 120 minutes / 6 minutes per mask = 20 masks.3. **Third and Fourth Hours:** - Manolo takes a 30-minute break. - During this break, he makes 0 masks.4. **Fourth to Sixth Hours:** - Manolo makes one face-mask every 8 minutes. - This period lasts for 2 hours, totaling 120 minutes. - Number of masks = 120 minutes / 8 minutes per mask = 15 masks.5. **Sixth to Eighth Hours:** - Manolo makes one face-mask every 6 minutes. - This period lasts for 2 hours, totaling 120 minutes. - Number of masks = 120 minutes / 6 minutes per mask = 20 masks.Finally, I'll sum up the number of masks made in each period to find the total number of face-masks Manolo produces during his eight-hour shift.

💡To determine the equation of the line passing through the point ( P(5, 6) ) with the x-intercept twice the y-intercept, we start by defining the intercepts. Let the y-intercept be ( b ). Then, the x-intercept is ( 2b ).The equation of the line in intercept form is:[frac{x}{2b} + frac{y}{b} = 1]Simplifying, we get:[frac{x}{2b} + frac{y}{b} = 1 implies x + 2y = 2b]Since the line passes through ( (5, 6) ), substituting these coordinates into the equation:[5 + 2(6) = 2b implies 5 + 12 = 2b implies 17 = 2b implies b = frac{17}{2}]Substituting ( b = frac{17}{2} ) back into the equation:[x + 2y = 17]Thus, the equation of the line is:[x + 2y - 17 = 0]Additionally, considering the possibility of the line passing through the origin, we assume the equation as ( y = mx ). Substituting ( (5, 6) ):[6 = 5m implies m = frac{6}{5}]Thus, the equation is:[6x - 5y = 0]Therefore, the final equations of the line are:[boxed{x + 2y - 17 = 0 text{ and } 6x - 5y = 0}]

💡First, I recognize that ( a^{m-n} ) can be expressed as ( frac{a^m}{a^n} ) using the properties of exponents.Given that ( a^m = 6 ) and ( a^{m-n} = 2 ), I substitute these values into the equation:[frac{6}{a^n} = 2]To solve for ( a^n ), I multiply both sides of the equation by ( a^n ):[6 = 2 cdot a^n]Next, I divide both sides by 2 to isolate ( a^n ):[a^n = frac{6}{2} = 3]Therefore, the value of ( a^n ) is 3.

💡First, I need to determine how much shampoo is left in the bottle after 4 days. Since the dad uses 1 ounce of shampoo each day, after 4 days, he has used 4 ounces. Therefore, there are 6 ounces of shampoo remaining in the 10-ounce bottle.Next, I know that after 4 days, 25% of the liquid in the bottle is hot sauce. This means that the total amount of liquid in the bottle is now 6 ounces of shampoo plus 4 times the amount of hot sauce added each day (since Adonis adds hot sauce every day for 4 days).I can set up an equation where the amount of hot sauce (4x) is equal to 25% of the total liquid in the bottle. This gives me the equation:4x = 0.25 * (6 + 4x)To solve for x, I'll first distribute the 0.25 on the right side:4x = 1.5 + xNext, I'll subtract x from both sides to isolate the variable:3x = 1.5Finally, I'll divide both sides by 3 to find the value of x:x = 0.5Therefore, Adonis adds 0.5 ounces of hot sauce to the bottle each day.

💡First, identify the initial number of books Mike had, which is 35.Next, determine the total number of books Mike has after buying more, which is 56.To find out how many books Mike bought at the yard sale, subtract the initial number of books from the total number of books.So, 56 minus 35 equals 21.Therefore, Mike bought 21 books at the yard sale.

💡Alright, so I've got this problem here about vectors. Let me read it again to make sure I understand."Given vectors a and b satisfy |a| = 4, |b| = 1, and |a + 2b| = 2√3, then the angle between vectors a and b is ______."Okay, so we have two vectors, a and b. The magnitude of a is 4, the magnitude of b is 1, and when we add a and twice b, the magnitude of that resultant vector is 2 times the square root of 3. We need to find the angle between a and b.Hmm, I remember that when dealing with vectors and their magnitudes, the dot product comes into play, especially when we're dealing with angles. The formula for the magnitude of the sum of two vectors involves the dot product. Let me recall that formula.If I have two vectors, say u and v, then the magnitude of their sum, |u + v|, can be found using the formula:|u + v|² = |u|² + |v|² + 2(u · v)Right, so that's the formula. It comes from expanding the dot product of (u + v) with itself.In this problem, instead of u and v, we have a and 2b. So, let's apply the same formula here.Given |a + 2b| = 2√3, so squaring both sides gives:|a + 2b|² = (2√3)² = 4 * 3 = 12So, |a + 2b|² = 12.Now, using the formula:|a + 2b|² = |a|² + |2b|² + 2(a · 2b)Wait, hold on. Let me make sure I get this right. The formula is |u + v|² = |u|² + |v|² + 2(u · v). So in this case, u is a and v is 2b.So, |a + 2b|² = |a|² + |2b|² + 2(a · 2b)But actually, wait, that's not quite right. The formula is |u + v|² = |u|² + |v|² + 2(u · v). So, u is a, v is 2b, so |a + 2b|² = |a|² + |2b|² + 2(a · 2b)But actually, let's think about it again. The formula is |u + v|² = |u|² + |v|² + 2(u · v). So, if u is a and v is 2b, then it's |a|² + |2b|² + 2(a · 2b). Wait, but that seems a bit off because the dot product is linear, so a · 2b is 2(a · b). So, perhaps it's better to factor out the scalar first.Alternatively, maybe it's simpler to write it as:|a + 2b|² = |a|² + 4|b|² + 4(a · b)Because (2b) · (2b) is 4|b|², and 2(a · 2b) is 4(a · b). Wait, no, that might not be accurate. Let me double-check.Wait, no, actually, the formula is |a + 2b|² = |a|² + |2b|² + 2(a · 2b). So, |2b|² is 4|b|², and 2(a · 2b) is 4(a · b). So, putting it all together:|a + 2b|² = |a|² + 4|b|² + 4(a · b)Yes, that seems correct.Given that, we can plug in the values we know.We know |a| = 4, so |a|² = 16.We know |b| = 1, so |2b|² = 4*1 = 4.And |a + 2b|² = 12.So, substituting these into the equation:12 = 16 + 4 + 4(a · b)Simplify the right side:16 + 4 = 20, so:12 = 20 + 4(a · b)Now, let's solve for (a · b):Subtract 20 from both sides:12 - 20 = 4(a · b)-8 = 4(a · b)Divide both sides by 4:-8 / 4 = a · b-2 = a · bSo, the dot product of a and b is -2.Now, the dot product is also equal to |a||b|cosθ, where θ is the angle between the vectors.So, a · b = |a||b|cosθWe know a · b = -2, |a| = 4, |b| = 1.So, substituting:-2 = 4 * 1 * cosθSimplify:-2 = 4cosθDivide both sides by 4:-2 / 4 = cosθ-0.5 = cosθSo, cosθ = -0.5Now, we need to find the angle θ where the cosine is -0.5.I remember that cos(120°) = -0.5, and cos(2π/3) = -0.5 in radians.Since angles between vectors are typically measured between 0 and π radians (or 0° and 180°), 120° or 2π/3 radians is the correct angle.So, θ = 2π/3 radians.Let me just recap to make sure I didn't make any mistakes.1. Start with |a + 2b| = 2√3, so square both sides to get |a + 2b|² = 12.2. Use the formula |a + 2b|² = |a|² + |2b|² + 4(a · b).3. Plug in |a| = 4, |b| = 1, so |a|² = 16, |2b|² = 4.4. So, 12 = 16 + 4 + 4(a · b) => 12 = 20 + 4(a · b) => 4(a · b) = -8 => a · b = -2.5. Then, a · b = |a||b|cosθ => -2 = 4*1*cosθ => cosθ = -0.5.6. Therefore, θ = arccos(-0.5) = 120° or 2π/3 radians.Yes, that all checks out. I think that's the correct answer.**Final Answer**The angle between vectors overrightarrow{a} and overrightarrow{b} is boxed{dfrac{2pi}{3}}.