The length and breadth of a cuboid are 70 cm and 96 cm, respectively. If its surface area is 25392 cm2, then what is the length (in cm) of its diagonal?
\(\sqrt{15402}\)
\(\sqrt{15420}\)
\(\sqrt{15408}\)
\(\sqrt{15412}\)
To find the length of the diagonal of a cuboid, we first need to determine its height. We are given the length, breadth, and surface area. We can use the formula for the total surface area of a cuboid to find the height. Once the height is known, we can use the formula for the diagonal of a cuboid.
The total surface area of a cuboid is given by:
\[ SA = 2(lb + bh + hl) \]Let \(h\) be the height of the cuboid.
\[ 25392 = 2((70)(96) + (96)h + h(70)) \]\[ 25392 = 2(6720 + 96h + 70h) \]\[ 25392 = 2(6720 + 166h) \]Divide both sides by 2:
\[ \frac{25392}{2} = 6720 + 166h \]\[ 12696 = 6720 + 166h \]\[ 166h = 12696 - 6720 \]\[ 166h = 5976 \]\[ h = \frac{5976}{166} \]\[ h = 36 \text{ cm} \]The length of the diagonal \((D)\) of a cuboid is given by:
\[ D = \sqrt{l^2 + b^2 + h^2} \]The length of the diagonal is \(\sqrt{15412}\) cm.
D) \(\sqrt{15412}\)
The calculation shows that the diagonal length is \(\sqrt{15412}\) cm, which matches option D.
The problem requires us to modify a given number based on specific rules for even and odd digits, and then find the sum of two specific digits in the new number.
Thus, the sum of the second digit from the left and the third digit from the right in the new number is 12.
Correct Option: D) 12
The problem requires us to modify a given number based on specific rules for its odd and even digits, then identify the highest and lowest digits in the new number, and finally calculate their sum.
D) 10 is the correct answer because, after applying the given rules, the new number formed is 2628464. The highest digit in this new number is 8, and the lowest digit is 2. Their sum is \(8 + 2 = 10\).
The problem asks us to identify the highest and lowest numbers from a given set of three-digit numbers, then extract specific digits from them and perform an addition operation. We need to find the second digit of the highest number and the third digit of the lowest number and add them together.
Result \( = 5 + 5 = 10 \)
A) 10 — As calculated, the sum of the second digit of the highest number (5) and the third digit of the lowest number (5) is 10.
We are given the selling price of an article as ₹x. We are also given a scenario where if the article is sold at 40% of this price, there is a 50% loss. We need to find the percentage profit when the article is sold for ₹x. We will first find the cost price (CP) using the information from the loss scenario, and then calculate the profit percentage when sold at ₹x.
Let the original selling price be \(SP_1 = \text{₹}x\).
According to the problem, if the article is sold at 40% of this price, there is a loss of 50%.
The new selling price \(SP_2\) is:
\[ SP_2 = 40\% \text{ of } x = \frac{40}{100} \times x = 0.4x \]When sold at \(SP_2\), there is a loss of 50%. The formula for selling price with loss is:
\[ SP = CP \times \left(1 - \frac{\text{Loss}\%}{100}\right) \]Substituting the values:
\[ 0.4x = CP \times \left(1 - \frac{50}{100}\right) \] \[ 0.4x = CP \times \left(1 - 0.5\right) \] \[ 0.4x = CP \times 0.5 \]Now, we can find the Cost Price (CP):
\[ CP = \frac{0.4x}{0.5} = \frac{4x}{5} = 0.8x \]We need to find the percentage profit when the article is sold for ₹x (which is \(SP_1\)).
Profit = \(SP_1 - CP\)
\[ \text{Profit} = x - 0.8x = 0.2x \]The percentage profit is calculated as:
\[ \text{Profit}\% = \frac{\text{Profit}}{CP} \times 100 \] \[ \text{Profit}\% = \frac{0.2x}{0.8x} \times 100 \] \[ \text{Profit}\% = \frac{0.2}{0.8} \times 100 \] \[ \text{Profit}\% = \frac{1}{4} \times 100 \] \[ \text{Profit}\% = 25\% \]Correct Option: A) 25%
The problem asks us to identify the letter-cluster that does not follow the same pattern as the others, based on their positions in the English alphabetical order. We need to find the positional values of each letter and then look for a consistent mathematical or sequential relationship within each cluster.
Let's assign numerical values to the letters in each option:
Differences:
So, the differences are \(+5, +4, +3\).
Differences:
So, the differences are \(+5, +4, +3\).
Differences:
This pattern is not \(+5, +4, +3\).
Differences:
So, the differences are \(+5, +4, +3\).
Options A, B, and D all follow the pattern where the difference between consecutive letters decreases by 1, starting from +5 (i.e., +5, +4, +3). Option C does not follow this pattern.
C) FYUB — This cluster does not follow the pattern of differences \(+5, +4, +3\) between consecutive letters, unlike the other three options.
If a + b = 32 and (a − b)2 = 372, find the value of product of a and b.
The problem provides two equations involving two variables, 'a' and 'b'. We need to find the product of 'a' and 'b'. We can use algebraic identities to solve this problem efficiently.
We know the identity:
\[ (a + b)^2 - (a - b)^2 = 4ab \]We are given:
Substitute these values into the identity:
\[ (32)^2 - 372 = 4ab \]Divide both sides by 4 to find the value of \(ab\):
\[ ab = \frac{652}{4} \]\[ ab = 163 \]Correct Option: B) 163
This problem involves a property of points inside a rectangle. For any point P inside a rectangle ABCD, the sum of the squares of the distances from P to opposite vertices is equal. That is, \(PA^2 + PC^2 = PB^2 + PD^2\). We can use this property to find the unknown length PD.
Correct Option: C) 98
This problem involves calculating the simple interest rate required for a sum of money to triple itself over a given period. We will use the simple interest formula to establish a relationship between the principal, amount, interest, time, and rate, and then solve for the unknown rate.
Correct Option: C) 25
The problem involves calculating an individual's investment in a partnership given the profit earned, the profit percentage of total investment, and the investment ratio among partners. We will first find the total investment using the profit and its percentage. Then, we will use the investment ratio to determine Naveen's share of the total investment.
Calculate the Total Investment:
The profit earned is ₹36,400, which is 14% of the total investment.
\[ \text{Profit} = 14\% \text{ of Total Investment} \]So,
\[ 36,400 = \frac{14}{100} \times \text{Total Investment} \]Rearranging the formula to find the Total Investment:
\[ \text{Total Investment} = \frac{36,400 \times 100}{14} \]\[ \text{Total Investment} = \frac{3,640,000}{14} \]\[ \text{Total Investment} = 260,000 \text{ ₹} \]Determine Naveen's Investment:
The investment ratio of Reena : Naveen : Arpita is 5 : 6 : 13.
The sum of the ratios is \(5 + 6 + 13 = 24\).
Naveen's share in the investment is \(\frac{6}{24}\) of the total investment.
\[ \text{Naveen's Investment} = \frac{6}{24} \times \text{Total Investment} \]\[ \text{Naveen's Investment} = \frac{1}{4} \times 260,000 \]\[ \text{Naveen's Investment} = 65,000 \text{ ₹} \]A) ₹65,000 is the correct investment made by Naveen, as calculated by first finding the total investment from the given profit and percentage, and then applying Naveen's share of the investment ratio.