💡First, I need to determine the interest rate that causes a difference of Rs. 65 between compound interest and simple interest over a 2-year period for a principal amount of Rs. 6,499.99.I'll start by recalling the formulas for simple interest (SI) and compound interest (CI). Simple interest is calculated as:[ SI = frac{P times r times t}{100} ]where ( P ) is the principal, ( r ) is the interest rate, and ( t ) is the time in years.Compound interest, compounded annually, is calculated as:[ CI = P times left(1 + frac{r}{100}right)^t - P ]Given that the difference between CI and SI is Rs. 65, I can set up the equation:[ CI - SI = 65 ]Substituting the formulas into this equation:[ P times left(1 + frac{r}{100}right)^2 - P - frac{P times r times 2}{100} = 65 ]Plugging in the given values for ( P ) and ( t ):[ 6,499.99 times left(1 + frac{r}{100}right)^2 - 6,499.99 - frac{6,499.99 times r times 2}{100} = 65 ]To simplify, I'll divide the entire equation by 6,499.99:[ left(1 + frac{r}{100}right)^2 - 1 - frac{2r}{100} = frac{65}{6,499.99} ]Expanding the left side:[ 1 + frac{2r}{100} + left(frac{r}{100}right)^2 - 1 - frac{2r}{100} = frac{65}{6,499.99} ]The ( frac{2r}{100} ) terms cancel out, leaving:[ left(frac{r}{100}right)^2 = frac{65}{6,499.99} ]Solving for ( r ):[ frac{r^2}{10,000} = frac{65}{6,499.99} ][ r^2 = left(frac{65}{6,499.99}right) times 10,000 ][ r^2 approx 1000 ][ r approx sqrt{1000} ][ r approx 31.62% ]Therefore, the interest rate is approximately 31.62% per annum.

💡Okay, so I need to find the equation of the directrix of the parabola given by the equation ( y = frac{x^2 - 4x + 4}{8} ). Hmm, I remember that the directrix is a line that's used in the definition of a parabola, right? It's the line that, together with the focus, defines the parabola as the set of all points equidistant from both the focus and the directrix.First, I think it would help to rewrite the given equation in a more standard form. The equation is ( y = frac{x^2 - 4x + 4}{8} ). I notice that the numerator is a quadratic expression, so maybe I can complete the square to put it into vertex form. Vertex form is usually ( y = a(x - h)^2 + k ), where ( (h, k) ) is the vertex of the parabola.Let me try completing the square for the numerator. The expression is ( x^2 - 4x + 4 ). I know that completing the square involves taking the coefficient of ( x ), which is -4, dividing it by 2 to get -2, and then squaring it to get 4. So, that means the expression can be written as ( (x - 2)^2 ). Let me check that: ( (x - 2)^2 = x^2 - 4x + 4 ). Yes, that's correct.So, substituting back into the original equation, we have:[ y = frac{(x - 2)^2}{8} ]Which simplifies to:[ y = frac{1}{8}(x - 2)^2 ]Now, this looks like the standard form of a parabola that opens upwards. The general form is ( y = a(x - h)^2 + k ), where ( (h, k) ) is the vertex. In this case, ( h = 2 ) and ( k = 0 ), so the vertex of the parabola is at ( (2, 0) ).I remember that for a parabola in the form ( y = a(x - h)^2 + k ), the focus is located at ( (h, k + frac{1}{4a}) ) and the directrix is the line ( y = k - frac{1}{4a} ). So, in this case, ( a = frac{1}{8} ), ( h = 2 ), and ( k = 0 ).Let me calculate the directrix. The formula for the directrix is ( y = k - frac{1}{4a} ). Plugging in the values, we get:[ y = 0 - frac{1}{4 times frac{1}{8}} ]Simplifying the denominator:[ 4 times frac{1}{8} = frac{4}{8} = frac{1}{2} ]So, the equation becomes:[ y = 0 - frac{1}{frac{1}{2}} ]Dividing by a fraction is the same as multiplying by its reciprocal, so:[ y = 0 - 2 ]Which simplifies to:[ y = -2 ]Wait, that doesn't seem right. I thought the directrix was closer to the vertex, but maybe I made a mistake in my calculation. Let me double-check.The formula for the directrix is ( y = k - frac{1}{4a} ). Here, ( a = frac{1}{8} ), so ( 4a = frac{4}{8} = frac{1}{2} ). Therefore, ( frac{1}{4a} = frac{1}{frac{1}{2}} = 2 ). So, subtracting that from ( k = 0 ), we get ( y = -2 ). Hmm, that seems correct.But wait, I recall that for a parabola ( y = ax^2 + bx + c ), the focal length ( p ) is given by ( p = frac{1}{4a} ). So, in this case, ( p = frac{1}{4 times frac{1}{8}} = frac{1}{frac{1}{2}} = 2 ). That means the focus is 2 units above the vertex, and the directrix is 2 units below the vertex.Since the vertex is at ( (2, 0) ), moving 2 units below would place the directrix at ( y = -2 ). That makes sense because the parabola opens upwards, so the directrix should be below the vertex.Let me visualize this. The vertex is at ( (2, 0) ), and the parabola opens upwards, so the focus is at ( (2, 2) ) and the directrix is the line ( y = -2 ). That seems consistent.I think I might have confused myself earlier when I thought the directrix was closer, but actually, since the coefficient ( a ) is ( frac{1}{8} ), which is less than 1, the parabola is wider, meaning the focal length is longer. So, a larger focal length means the directrix is further away from the vertex.To confirm, let's recall that for the standard parabola ( y = ax^2 ), the directrix is ( y = -frac{1}{4a} ). In our case, after completing the square, the equation is ( y = frac{1}{8}(x - 2)^2 ), so ( a = frac{1}{8} ). Therefore, the directrix should be ( y = -frac{1}{4 times frac{1}{8}} = -2 ). Yep, that checks out.So, after going through the steps, I'm confident that the equation of the directrix is ( y = -2 ).

💡First, I need to find the prime factors of 1764. I'll start by dividing the number by the smallest prime, which is 2.1764 divided by 2 equals 882. Since 882 is still even, I'll divide by 2 again to get 441.Now, 441 is odd, so I'll check the next prime number, which is 3. Dividing 441 by 3 gives me 147. I'll divide by 3 again to get 49.Next, I'll check if 49 is divisible by 3. It's not, so I'll move to the next prime number, which is 7. Dividing 49 by 7 gives me 7, and dividing by 7 again gives me 1.So, the prime factors of 1764 are 2, 3, and 7. Among these, the largest prime factor is 7.

💡First, I need to determine the number of each type of jar Jill has. She has quart jars, half-gallon jars, and one-gallon jars, and the number of each type is the same.Let’s denote the number of each type of jar as ( x ).- **Quart jars**: Each holds ( frac{1}{4} ) gallon.- **Half-gallon jars**: Each holds ( frac{1}{2} ) gallon.- **One-gallon jars**: Each holds ( 1 ) gallon.The total volume of water stored in all jars is 28 gallons. Therefore, the equation representing the total volume is:[frac{1}{4}x + frac{1}{2}x + x = 28]To combine these terms, I'll find a common denominator, which is 4:[frac{1}{4}x + frac{2}{4}x + frac{4}{4}x = 28]Adding the fractions:[frac{7}{4}x = 28]To solve for ( x ), I'll multiply both sides by ( frac{4}{7} ):[x = 28 times frac{4}{7} = 16]So, Jill has 16 jars of each type. The total number of jars is:[16 text{ (quart jars)} + 16 text{ (half-gallon jars)} + 16 text{ (one-gallon jars)} = 48 text{ jars}]

💡First, I need to determine the total amount Caleb spent on ice cream. He bought 10 cartons, and each carton costs 4. So, I multiply the number of cartons by the cost per carton to find the total cost for ice cream.Next, I calculate the total amount spent on frozen yogurt. Caleb purchased 4 cartons, and each carton costs 1. Again, I multiply the number of cartons by the cost per carton to find the total cost for frozen yogurt.Finally, to find out how much more was spent on ice cream compared to frozen yogurt, I subtract the total cost of frozen yogurt from the total cost of ice cream. This will give me the difference in spending between the two items.

💡Alright, let's try to figure out this problem step by step. So, Tom has 13 snakes in total. Out of these, 4 are purple, and 5 are happy. Now, there are some observations given:1. All of his happy snakes can add.2. None of his purple snakes can subtract.3. All of his snakes that can't subtract also can't add.We need to figure out which conclusion can be drawn from these statements. The options are:A) Purple snakes can add.B) Purple snakes are happy.C) Snakes that can add are purple.D) Happy snakes are not purple.E) Happy snakes can't subtract.Okay, let's break down each statement and see what they imply.First, statement 1: All happy snakes can add. So, if a snake is happy, it has the ability to add. That means the set of happy snakes is a subset of snakes that can add.Statement 2: None of the purple snakes can subtract. So, if a snake is purple, it cannot subtract. That means the set of purple snakes is a subset of snakes that cannot subtract.Statement 3: All snakes that can't subtract also can't add. So, if a snake cannot subtract, then it also cannot add. This means the set of snakes that can't subtract is a subset of snakes that can't add.Now, let's try to connect these statements. From statement 2, we know that purple snakes can't subtract. From statement 3, since they can't subtract, they also can't add. So, purple snakes can't add.From statement 1, happy snakes can add. So, if a snake is happy, it can add. But we just found out that purple snakes can't add. Therefore, happy snakes can't be purple because if they were, they wouldn't be able to add, which contradicts statement 1.So, putting it all together:- Happy snakes can add (statement 1).- Purple snakes can't subtract (statement 2).- Snakes that can't subtract can't add (statement 3).- Therefore, purple snakes can't add.- Since happy snakes can add, they can't be purple.This leads us to conclusion D: Happy snakes are not purple.Let's quickly check the other options to make sure we're not missing anything.A) Purple snakes can add. - We just concluded they can't, so this is false.B) Purple snakes are happy. - If they were happy, they could add, but we know they can't, so this is false.C) Snakes that can add are purple. - We know happy snakes can add, and they aren't purple, so this is false.E) Happy snakes can't subtract. - We don't have information about whether happy snakes can subtract or not. They can add, but the statements don't specify about subtracting. So, we can't conclude this.Therefore, the only valid conclusion is D.**Final Answer**boxed{D}