Simplify sqrt(27) x sqrt(15)

Question

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Here are the solutions to the problems:

: Simplify $\sqrt{27} \times \sqrt{15}$

Simplify each square root.
$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$ $\sqrt{15} = \sqrt{3 \times 5}$
Multiply the simplified terms.
$\sqrt{27} \times \sqrt{15} = 3\sqrt{3} \times \sqrt{3 \times 5}$ $= 3 \times \sqrt{3 \times 3 \times 5}$ $= 3 \times \sqrt{9 \times 5}$ $= 3 \times \sqrt{9} \times \sqrt{5}$ $= 3 \times 3 \times \sqrt{5}$ $= 9\sqrt{5}$
The correct option is D.
$\boxed{\text{D. } 9\sqrt{5}}$
: Find the radius of a circle whose area is the same as that of a rectangle of sides 4cm by 11cm. (take $\pi = \frac{22}{7}$ )
Calculate the area of the rectangle.
$\text{Area of rectangle} = \text{length} \times \text{width} = 11 \text{ cm} \times 4 \text{ cm} = 44 \text{ cm}^2$
Set the area of the circle equal to the area of the rectangle.
$\text{Area of circle} = \pi r^2$ $\pi r^2 = 44 \text{ cm}^2$
Substitute the value of $\pi$ and solve for $r$ .
$\frac{22}{7} r^2 = 44$ $r^2 = 44 \times \frac{7}{22}$ $r^2 = 2 \times 7$ $r^2 = 14$ $r = \sqrt{14} \text{ cm}$
Approximate the value of $r$ and compare with options.
$\sqrt{14} \approx 3.74 \text{ cm}$
The closest option is B.
$\boxed{\text{B. } 3.7 \text{ cm}}$
: Find an expression for the nth term, $U_n$ , of the sequence -2, 3, -4, 5, ...
Analyze the sequence.
The absolute values of the terms are 2, 3, 4, 5, ..., which corresponds to $(n+1)$ for $n=1, 2, 3, 4, ...$ .
The signs alternate: negative, positive, negative, positive, ...
For $n=1$ , the term is negative. For $n=2$ , the term is positive.
This pattern of signs can be represented by $(-1)^n$ or $(-1)^{n+1}$ .
If $n=1$ , $(-1)^1 = -1$ (negative).
If $n=2$ , $(-1)^2 = 1$ (positive).
So, the sign factor is $(-1)^n$ .
Combine the absolute value and sign factor.
The nth term $U_n$ is $(-1)^n \times (n+1)$ .

The correct option is C.
$\boxed{\text{C. } (-1)^n(n+1)}$
: Find the area of an equilateral triangle whose sides are 6cm.
Recall the formula for the area of an equilateral triangle.
The area ( $A$ ) of an equilateral triangle with side length $s$ is given by:
$A = \frac{\sqrt{3}}{4} s^2$
Substitute the given side length $s = 6$ cm.
$A = \frac{\sqrt{3}}{4} (6 \text{ cm})^2$ $A = \frac{\sqrt{3}}{4} \times 36 \text{ cm}^2$ $A = 9\sqrt{3} \text{ cm}^2$
The correct option is D.
$\boxed{\text{D. } 9\sqrt{3} \text{ cm}^2}$
: How many terms are in an Arithmetic Progression whose third term is 6 and the common difference and last term are 2 and 100 respectively?
Identify the given information.
Third term ( $U_3$ ) = 6
Common difference ( $d$ ) = 2
Last term ( $U_n$ ) = 100
Use the formula for the nth term of an AP, $U_n = a + (n-1)d$ , to find the first term ( $a$ ).
For the third term:
$U_3 = a + (3-1)d$ $6 = a + 2(2)$ $6 = a + 4$ $a = 6 - 4$ $a = 2$
Use the first term, common difference, and last term to find the number of terms ( $n$ ).
$U_n = a + (n-1)d$ $100 = 2 + (n-1)2$ $100 - 2 = (n-1)2$ $98 = 2(n-1)$ $\frac{98}{2} = n-1$ $49 = n-1$ $n = 49 + 1$ $n = 50$
The correct option is A.
$\boxed{\text{A. } 50}$
: Find the perimeter of a parallelogram whose adjacent sides are of lengths 8cm and 12cm.
Recall the formula for the perimeter of a parallelogram.
A parallelogram has two pairs of equal adjacent sides. If the lengths of the adjacent sides are $a$ and $b$ , the perimeter ( $P$ ) is given by:
$P = 2(a+b)$
Substitute the given side lengths.
Given $a = 8$ cm and $b = 12$ cm.
$P = 2(8 \text{ cm} + 12 \text{ cm})$ $P = 2(20 \text{ cm})$ $P = 40 \text{ cm}$
The correct option is D.
$\boxed{\text{D. } 40 \text{ cm}}$
: Let P and Q represent the statements: P = Olu is brilliant, Q = Sola is lazy. Which of the following does p $\land$ q represent?
Understand the logical connective $\land$ .
The symbol $\land$ represents the logical operator "AND".
Combine the statements P and Q using "AND".
P: Olu is brilliant
Q: Sola is lazy
p $\land$ q means "Olu is brilliant AND Sola is lazy".

The correct option is C.
$\boxed{\text{C. Olu is brilliant and Sola is lazy}}$

Simplify sqrt(27) x sqrt(15)

Answer

Step-by-step solution

Simplify each square root.

Multiply the simplified terms.

Calculate the area of the rectangle.

Set the area of the circle equal to the area of the rectangle.

Substitute the value of $\pi$ and solve for $r$ .

Approximate the value of $r$ and compare with options.

Analyze the sequence.

Combine the absolute value and sign factor.

Recall the formula for the area of an equilateral triangle.

Substitute the given side length $s = 6$ cm.

Identify the given information.

Use the formula for the nth term of an AP, $U_n = a + (n-1)d$ , to find the first term ( $a$ ).

Use the first term, common difference, and last term to find the number of terms ( $n$ ).

Recall the formula for the perimeter of a parallelogram.

Substitute the given side lengths.

Understand the logical connective $\land$ .

Combine the statements P and Q using "AND".

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify sqrt(27) x sqrt(15)

Answer

Step-by-step solution

Simplify each square root.

Multiply the simplified terms.

Calculate the area of the rectangle.

Set the area of the circle equal to the area of the rectangle.

Substitute the value of π\piπ and solve for rrr.

Approximate the value of rrr and compare with options.

Analyze the sequence.

Combine the absolute value and sign factor.

Recall the formula for the area of an equilateral triangle.

Substitute the given side length s=6s = 6s=6 cm.

Identify the given information.

Use the formula for the nth term of an AP, Un=a+(n−1)dU_n = a + (n-1)dUn​=a+(n−1)d, to find the first term (aaa).

Use the first term, common difference, and last term to find the number of terms (nnn).

Recall the formula for the perimeter of a parallelogram.

Substitute the given side lengths.

Understand the logical connective ∧\land∧.

Combine the statements P and Q using "AND".

Try one yourself

Still stuck on this one?

More Mathematics Questions

Substitute the value of $\pi$ and solve for $r$ .

Approximate the value of $r$ and compare with options.

Substitute the given side length $s = 6$ cm.

Use the formula for the nth term of an AP, $U_n = a + (n-1)d$ , to find the first term ( $a$ ).

Use the first term, common difference, and last term to find the number of terms ( $n$ ).

Understand the logical connective $\land$ .