The sum of two numbers is 66. the larger number is 2 more than 3 times the smaller. what are the two numbers?

Final answer:

To use properties of exponents and logarithms effectively, one must apply specific rules such as the logarithm of a product being the sum of the logarithms and the logarithm of a number raised to an exponent being the product of the exponent and the logarithm. Log transformations help linearize and solve equations involving exponents.

Explanation:

To rewrite functions in equivalent forms and solve equations using the properties of exponents and logarithms, one must understand several key rules and relationships. For example, the common logarithm (log) indicates the power to which 10 must be raised to equal a given number.

Two essential properties we use are:

Exponential and logarithmic functions are inverses. This relationship allows us to simplify complex expressions and solve exponential equations by taking the logarithm of both sides. For instance, the natural logarithm (ln or loge) and the base e exponential are inverse functions, so ln (e^x) = x and e^(ln x) = x.

A log transformation can linearize functions that involve an exponent by taking the log of both sides of the equation. This technique is valuable in solving equations where the variable is an exponent.

The student's conversion calculation of speed units from kilometers per minute to miles per hour yields a speed of 2.4 miles/hour, so the answer is choice A.

The question is a conversion exercise in speed units from kilometers per minute to miles per hour. We have the following information: 3.2 km in 50 min, and we know that 1 mile is equivalent to 1.6 km and 60 min is equivalent to 1 hour. Thus, the conversion is as follows:

3.2 km ÷ 50 min. = 0.064 km/min.

Converting km/min to mi/hr involves multiplying by 60 (minutes in an hour) and dividing by 1.6 (km in a mile), and we get:

(0.064 km/min x 60 min/hr) ÷ 1.6 km/mi. = 2.4 mi/hr.

Hence, the correct answer is (A) 2.4 miles/hour.

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Answer:

2b - 3 = 5

Step-by-step explanation:

If her brother read b number of books last summer, twice the number of books her brother read

= 2b

three less than twice as many books as her brother read

= 2b - 3

If haley read five books and it is same as three less than twice as many books as her brother last summer, then

2b - 3 = 5

Answer with Step-by-step explanation:

As we can see the venn diagram:

number of bird+fish=6

number of bird+not fish=8

number of fish+not bird=19

and number of not fish and not bird=22

Hence, we get the following table

                        Fish           Not Fish            Total

Bird                6                    8                       14

Not Bird         19                  22                       41

Total               25                 30                       55

                Fish    Not Fish   Total

Bird            6            18           14

Not Bird    19           22          41

Total          25           30         55

Plato explains that we know geometry by our gain knowledge through recollection. Our soul is what recollects this place hence we came where there exist unchanging truths. Delivered the theory of Forms, according to which the world people know by means of the senses is just an imitation of the eternal, pure, eternal, and fixed world of the Forms.

Answer:

Option B is correct.

add 4 times the second equation to 3 times the first equation

Step-by-step explanation:

Given the system of equation:

[tex]2x - 4y = 6[/tex]                         ......[1]

[tex]-3x + 3y =12[/tex]                       .....[2]

Multiply equation [1] by 3 we get;

[tex]3(2x-4y) = 3 \cdot 6[/tex]

Using distributive property; [tex]a\cdot (b+c) = a\cdot b+ a\cdot c[/tex]

6x - 12y = 18                                   .......[3]

Multiply equation [2] by 4 we get;

[tex]4(-3x+3y) = 4 \cdot 12[/tex]

Using distributive property we get;

-12x + 12y = 48                                   ......[4]

Add equation [3] and [4] to eliminate y and solve for x;

(6x -12y) + ( -12x +12y ) = 18 + 48

6x - 12y -12x + 12y = 66

Combine like terms;

6x - 12x = 66

or

-6x = 66

Simplify:

x = -11

Therefore, the operation which could be used to eliminate the y-variable and find the value of x is;

add 4 times the second equation to 3 times the first equation.

Answer:

Option B

Step-by-step explanation:

Given is a system of two equations as

[tex]2x − 4y = 6 \\−3x + 3y = 12[/tex]

To solve them using elimination:

Elimination method is the method of making coefficients of one variable numerically equal

IN the given system we have -4 as coefficient for y in I equation and 3 for y in II equation.

To make them numerically equal with opposite signs we can multiply I equation by 3 and II by 4

WE get

[tex]6x-12y =18\\-12x+12y=48[/tex]

Now if we add these two we are able to eliminate y from the system

Adding gives

[tex]-6x=66\\x=-11[/tex]

So option B is right

The height of cylinder is,

⇒ h = 2.4 inches

We have to given that,

In a cylinder,

Volume = 271.4 inches³

And, Radius = 6 inch

Since, We know that,

Volume of cylinder is,

⇒ V = πr²h

Where, V is volume , r is radius and h is height.

Substitute all the values, we get;

⇒ 271.4 = 3.14 × 6² × h

⇒ 271.4 = 3.14 × 36 × h

⇒ 271.4 = 113.04 × h

⇒ h = 2.4 inches

Thus, The height of cylinder is,

⇒ h = 2.4 inches

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We have been given that the chords AB and CD are equal.

Now, we know that two equal chords subtends equal angle at the center of the circle.

The chord CD subtends and angle CPD at the center and the chord AB subtends an angle APB at the center.

Hence, from above mentioned theorem, we can say that

[tex]\angle CPD = \angle APB[/tex]

Now, we have been given that

[tex]\angle CPD =50 ^{\circ} [/tex]

[tex]\angle APB =x ^{\circ} [/tex]

Hence, we have

[tex]x=50^{\circ}[/tex]

Answer:

The equation is [tex]y ^ 2 = \frac{x + 1}{2}[/tex]

Step-by-step explanation:

The parameter that we have is t. We want to eliminate this parameter in both equations, therefore in the first equation we solve for t and in the second equation we solve for the variable t.

We have:

[tex]x = 2t-1\\\\x + 1 = 2t\\\\t = \frac{x + 1}{2}[/tex]

Now we solve the other equation for t.

[tex]y= \sqrt{t}[/tex]

[tex]y ^ 2 = t[/tex]     because   [tex]t> 0[/tex]

As [tex]t = y ^ 2[/tex] and also [tex]t = \frac{x + 1}{2}[/tex]

Then:

[tex]y ^ 2 = \frac{x + 1}{2}[/tex]

The correct answer is:

D. [tex]x=2y^{2} -1, y\geq 0[/tex]