Solve each equation (quadratic pattern)

[tex]2^{2x} -2^{x} =12[/tex]

[tex]3^{2x} + 3^{x+1} =4[/tex]

[tex]4^{x} + 6[/tex] · [tex]2^{x} +8 = 0[/tex]

[tex]9^{x} = 3^{x} +6[/tex]

Answer:

Part A) The cone couldn't contain all the ice cream if it melted.

Part B) The height of the cone would be [tex]8\ cm[/tex]

Part C) The height of the cylinder would be [tex]3\ cm[/tex]

Step-by-step explanation:

Part A) Determine whether the cone could contain all of the ice cream if it melted

step 1

Find the volume of the ice cream (sphere)

The volume is equal to

[tex]V=\frac{4}{3}\pi r^{3}[/tex]

we have

[tex]r=4/2=2\ cm[/tex] -----> the radius is half the diameter

substitute

[tex]V=\frac{4}{3}\pi (2)^{3}=\frac{32}{3}\pi\ cm^{3}[/tex]

step 2

Find the volume of the cone

The volume is equal to

[tex]V=\frac{1}{3}\pi r^{2}h[/tex]

we have

[tex]r=4/2=2\ cm[/tex] -----> the radius is half the diameter

[tex]h=7.5\ cm[/tex]

substitute

[tex]V=\frac{1}{3}\pi (2)^{2}(7.5)=\frac{30}{3}\pi\ cm^{3}[/tex]

step 3

Compare the volume of the sphere and the volume of the cone

[tex]\frac{30}{3}\pi\ cm^{3} < \frac{32}{3}\pi\ cm^{3}[/tex]

The volume of the cone is less than the volume of the sphere

therefore

The cone couldn't contain all the ice cream if it melted.

Part B) What would be the smallest cone in height in whole centimeters that would allow the cone to contain all of the melted ice cream if the diameter of the cone remains unchanged

The volume of the cone is equal to

[tex]V=\frac{1}{3}\pi r^{2}h[/tex]

we have

[tex]V=\frac{32}{3}\pi\ cm^{3}[/tex]

[tex]r=2\ cm[/tex]

substitute in the formula and solve for h

[tex]\frac{32}{3}\pi=\frac{1}{3}\pi (2)^{2}h[/tex]

simplify

[tex]32=(2)^{2}h[/tex]

[tex]32=4h[/tex]

[tex]h=32/4=8\ cm[/tex]

Part C) If the container of the ice cream changed to a cylinder as shown in the diagram below, what would be the smallest height of the cylinder needed to the nearest whole centimeter to contain the melted ice cream

The volume of the cylinder is equal to

[tex]V=\pi r^{2}h[/tex]

we have

[tex]V=\frac{32}{3}\pi\ cm^{3}[/tex]

[tex]r=2\ cm[/tex]

substitute in the formula and solve for h

[tex]\frac{32}{3}\pi=\pi (2)^{2}h[/tex]

simplify

[tex]\frac{32}{3}=(2)^{2}h[/tex]

[tex]\frac{32}{3}=4h[/tex]

[tex]h=\frac{32}{12}=2.67\ cm[/tex]

Round to the nearest whole centimeter

[tex]2.67=3\ cm[/tex]

Answer:

Option b

Step-by-step explanation:

We have a compound interest problem. With an annual interest rate of 0.675 and an initial payment of 8500, with t = 25 years

Then you must use the annual compound interest formula, which is represented by a growing exponential function:

[tex]y = e ^{ht}[/tex]

Where:

h is the interest rate of 0.675

y is the money in the savings account as a function of time

Then substitute the values in the formula and we have:

[tex]y = e ^{0.675(25)}[/tex]

[tex]y = 45,950.57[/tex]

Answer:

see explanation

Step-by-step explanation:

Divide through by 2

2a² - 5a + 3 = 0

To factor the quadratic

Consider the factors of the product of the coefficient of the a² term and the constant term which sum to give the coefficient of the x- term

product = 2 × 3 = 6 and sum = - 5

The factors are - 2 and - 3

Use the factors to split the a- term

2a² - 2a - 3a + 3 = 0 ( factor the first/second and third/fourth terms )

2a(a - 1) - 3(a - 1) = 0 ← factor out (a - 1)

(a - 1)(2a - 3) = 0

Equate each factor to zero and solve for a

a - 1 = 0 ⇒ a = 1

2a - 3 = 0 ⇒ 2a = 3 ⇒ a = [tex]\frac{3}{2}[/tex]

Answer:

a = 3/2 or 1

Step-by-step explanation:

4a²-10a+6=0

(Divide by 2)

2a²-5a+3=0

(Now factorise)

(2a-3)(a-1)

a = 3/2 or 1

Answer:

(x,y) = (0,-8)

Step-by-step explanation:

We know that a point in polar coordinates is represented by

(r, θ)

Where r is the distance from the origin and θ is the angle.

Rectangular coordinates can be found by

x = r*cos(θ)

y = r*sin (θ)

x = r*cos(θ) = 8* cos (3π/2)

y = r*sin (θ) = 8 sin(3π/2)

x = 8* cos (3π/2) = 8*0 = 0

y = 8* sin (3π/2) = 8*(-1) = -8

(x,y) = (0,-8)

Correct option is (A) (0,-8)

Now any point in polar coordinates is represented by

(r, θ)

where 'r' is the distance from the origin

and 'α' is the angle.

Rectangular coordinates can be found by using the formula:

[tex]x=r*cos(\alpha )\\y=r*sin(\alpha )[/tex]

Thus the x coordinate would be given as :

[tex]x=r*cos(\alpha )\\x=8*cos(\frac{3\pi }{2} )\\x=8*0\\x=0\\[/tex]

Similarly the y coordinate would be given as :

[tex]y=r*sin(\alpha )\\y=8*sin(\frac{3\pi }{2} )\\y=8*(-1)\\y=-8\\[/tex]

Thus (x, y) = (0,-8) is the required coordinates

Answer:

john

Step-by-step explanation:

a. 3u + 2v

To solve this problem, we need to apply some properties of logarithms. Properties are useful to simplify complicated expressions. Here we need to use a very useful property of logarithms called  the logarithm of a product is the sum of the logarithms, that is:

[tex]log_{b}(MN)=log_{b}(M)+log_{b}(N)[/tex]

From the function, it is then true that:

[tex]ln(x^{3}y^{2})=ln(x^{3})+ln(y^{2})[/tex]

The other property we must use is Logarithm of a Power:

[tex]log_{b}M^{n}=nlog_{b}M[/tex]

Then:

[tex]ln(x^{3}y^{2})=ln(x^{3})+ln(y^{2}) \\ \\ ln(x^{3}y^{2})=3ln(x)+2ln(y)[/tex]

Since:

[tex]u=ln(x) \\ v=ln(y)[/tex]

Then:

[tex]ln(x^{3}y^{2})=3u+2v[/tex]

Finally, the correct option is:

a. 3u + 2v

Answer:

A edge

Step-by-step explanation:

u didnt ask a question here you dummy thicc sassy block of cheesy

Manager 1, with 7 years of service, achieved an average daily sales of $5,000, maintained a high customer service rating of 5, and successfully completed 83% of projects on time Business involves the production, exchange, or provision of goods and services in pursuit of profit, contributing to economic growth and sustainability.

Manager 1's performance is evaluated based on several key metrics. First, their 7 years of service indicate experience and commitment to the company. Second, the daily sales average of $5,000 reflects their effectiveness in generating revenue. The customer service rating of 5 signifies exceptional customer satisfaction, indicating effective communication and problem-solving skills. Lastly, the 83% on-time project completion rate demonstrates efficiency in managing tasks and meeting deadlines. These attributes collectively suggest that Manager 1 is a valuable asset to the company, with a strong track record of delivering results and maintaining high standards of customer service.

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The smallest positive integer [tex]a[/tex] greater than 1000 such that [tex]\sqrt{a} - \sqrt{a-x}[/tex] has a rational root is [tex]a = 1024[/tex].

To find the smallest positive integer [tex]a[/tex] greater than 1000 such that the equation [tex]\sqrt{a} - \sqrt{a-x} = 0[/tex] has a rational root, we need to analyze the condition given in the equation.  

We start from the original equation:
[tex]\sqrt{a} - \sqrt{a-x} = 0[/tex]
This implies that:
[tex]\sqrt{a} = \sqrt{a-x}[/tex]

Squaring both sides will remove the square root:
[tex]a = a - x[/tex]
So, we simplify this to:
[tex]x = 0[/tex]

In this case, it suggests that for the equation to have a rational root, [tex]x[/tex] must be equal to zero, which is a trivial case and not within the scope of finding a number greater than 1000.  

Next, we need to find conditions under which [tex]\sqrt{a} - \sqrt{a-x}[/tex] has non-trivial rational roots. We realize that for other values of [tex]x[/tex], the right-hand side requires that [tex]a - x[/tex] must also be a perfect square in order for [tex]\sqrt{a-x}[/tex] to yield a rational number.  

Assume [tex]a = n^2[/tex] where [tex]n[/tex] is any integer. Therefore:
[tex]\sqrt{a} = n[/tex]
Then we rewrite the original equation in terms of a new term, say [tex]m[/tex], where:
[tex]a - x = m^2[/tex]
Substituting this, we find that:
[tex]n^2 - m^2 = x[/tex]

This indicates that [tex]x[/tex] must also be a perfect square if we want to maintain the rationality in all cases.  

We need to follow this procedure to find the smallest positive integer greater than 1000:  

While checking values yields rational results, the lowest value of [tex]a[/tex] that successfully gives a rational root while being above 1000 appears to be [tex]1024[/tex].

Final answer:

To find the percent of decrease, calculate the difference between the original and final numbers, divide by the original number, and multiply by 100. In this case, the percent of decrease is 25%.

Explanation:

To find the percent of decrease from the original number to the final number, we need to calculate the difference between the original number and the final number, then divide that difference by the original number and multiply by 100 to get the percentage.

Let's assume the original number is x. When the number is increased by 50%, it becomes 1.5x. When the result is decreased by 50%, it becomes 0.5 times 1.5x, which is 0.75x.

The decrease from the original number to the final number is x - 0.75x = 0.25x. To find the percent of decrease, we divide the decrease by the original number (0.25x / x) and multiply by 100 to get 25%. Therefore, the percent of decrease from the original number to the final number is 25%.

Final answer:

The overall percent change from the original number to the final number, after increasing by 50% and then decreasing by 50%, is a 25% decrease.

Explanation:

To find the percentage decrease from the original number to the final one after the series of increases and decreases described, we need to follow a couple of steps:

First, we increase the original number by 50%. If the original number is x, its increased value is x + 0.5x = 1.5x.

Next, we decrease this new number by 50%. The decreased value is 1.5x - (0.5 × 1.5x) = 1.5x - 0.75x = 0.75x.

To find the percentage change from the original value, we calculate (final value - initial value) / initial value × 100%. Using the value obtained from the second step, this becomes (0.75x - x) / x × 100% = -0.25x/x × 100% = -25%.

Therefore, the overall percent change is a 25% decrease from the original number.

Answer:

Step-by-step explanation:

d.3+6+9+12+15+18  sum 6 terms multiple of 3

For this question we need to use the Pythagorean Theorem (a2+b2=c2) since the rectangle is being divided into two triangles.

we know the length of the triangle (63) and we know the hypotenuse (65) but not the width. To find the width, we can plug the values we know into our formula.

(63) squared + b squared = (65) squared

solve.

3969 + b squared = 4225

subtract 3969 from both sides.

b squared = 256

√b² = √256

b=16

170.28 is the answer

Answer:

170.28

Step-by-step explanation:

Got it right on the test.

first blank- zero (0)

second balnk- the prime number

Answer:A prime number is a whole number greater than one whose only factors are 1 and itself

Answer:

50% chance

Step-by-step explanation:

You would have a 50% chance of landing on a even number

the correct answer would be 46°

It would be 46° but since there is an x = ? The "?" would be replaced with 46°

I hope this helps! ^^ You can just put 46° For your answer.

Answer:

Step-by-step explanation:

12 is the common denominator

The answer will be 6!!

The length of the missing side is √445 meters.

For a right triangle, the Pythagorean theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the legs).

In this case, we are given the lengths of the two legs, which are 11 meters and 18 meters. We need to find the length of the hypotenuse, which is the missing side.

Steps to solve:

Step 1: Substitute the given values into the Pythagorean theorem:

[tex]a^2 + b^2 = c^2[/tex]

where:

a = 11 meters (shorter leg)

b = 18 meters (longer leg)

c = the missing side (hypotenuse)

Step 2: Evaluate the equation:

[tex]11^2 + 18^2[/tex]= [tex]c^2[/tex]

121 + 324 = [tex]c^2[/tex]

445 = [tex]c^2[/tex]

Step 3: Take the square root of both sides to solve for c:

c = √445

The length of the missing side is √445 meters

Answer:

C

Step-by-step explanation:

We can use 2 properties of logarithms to write this:

1. ln(x*y) = lnx + ln y

2. ln(a^b) = b ln a

Using property 1, we can write as:

[tex]ln(\sqrt{x} *y^{2})\\=ln(\sqrt{x} )+ln(y^2)\\=ln(x^{\frac{1}{2}})+2lny\\=\frac{1}{2}lnx+2lny[/tex]

We know u = lnx and v = ln y, we simply substitute it now:

[tex]\frac{1}{2}lnx+2lny\\=\frac{1}{2}u+2v[/tex]

the correct answer is C

Answer:

c

Step-by-step explanation:

The expression is equivalent to (5x-6) (2x) + (5x-6) (3).

Option (A) is correct.

It is to find equivalent of (5x-6) (2x+3).

An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable.

when we multiply to (5x-6) to (2x+3).

Each term of (5x-6) is multiply to (2x+3).

so (5x-6) is multiply to 2x and (5x-6)  is multiply to 3 separately.

so, the expression is equivalent to (5x-6) (2x) + (5x-6) (3).

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Answer:

If Q=2.1(R)+5, then it would be 2.1(5)+5=15.5 ?

Step-by-step explanation:

Your horizontal value (x) is 3 because 5.5-2.5 = 3

Your vertical value (y) is 3.5 because 4.5-1 = 3.5

Pythagoras Theorem

[tex]c^{2}= \sqrt{a^{2}+b^{2}  }[/tex]

a = 3

b = 3.5

[tex]c^{2}= \sqrt{3^{2}+3.5^{2}  }[/tex]

c = 4.61 to 2d.p.

3/4 *40 =30.

So 30 cars have in state license

To find how many cars have in-state license plates, multiply the total number of cars (40) by the fraction (3/4). The answer is 30 cars have in-state license plates.

To find the number of cars with in-state license plates, we need to calculate 3/4 of the total number of cars in the parking lot.

Therefore, 30 cars in the parking lot have in-state license plates.

Correct question :

The parking lot has 40 cars and 3/4 of the cars have in-state license plates. How many cars have in-state license plates?

Answer:

it took a total of 260 minutes or 4 hours and 20 minutes from Collin's house to his hotel.

Step-by-step explanation:

As each of the activities described is an independent activity that does not overlap, we can easily sum up the durations of each to find the total time Collin took from his house to the hotel.

We add it as follows :

he drove to the airport for 45 minutes + he waited at the airport for the flight to depart for 90 minutes + his fight duration was 70 minutes + upon landing, he gathered his luggage for 20 minutes + he drove to the hotel for 35 minutes.

So, 45+90+70+20+35 = 260 minutes

Answer:

The numbers can be added in any order.

Step-by-step explanation:

Just got this quiz question right.

Hope this helps :)

The salary of the real estate broker = $18,000

Commission earned on total sales = 4% or 0.04

Total income earnings = $55,000

Let the total sales be = x

Equation becomes :

[tex]18000+0.04x=55000[/tex]

[tex]0.04x=55000-18000[/tex]

[tex]0.04x=37000[/tex]

[tex]x=925000[/tex]

Hence, the real estate broker must sell $925,000 worth of real estate to earn $55,000.

In the equation x is the feet of width.

If the original width is 2 feet, then X^2 = 2^2 = 4

If the width changes to 4 feet, then x^2 becomes 4^2 = 16

The change is 16 - 4 = 12

The answer should be B. $12 per foot.

Answer:

Option B is correct.

Step-by-step explanation:

Given the function which represent the total cost in dollars to build a boxed garden that is x feet wide.

[tex]C(x)=2x^2+32[/tex]

we have to find the average rate of change in the total cost to build the boxed garden as the width increases from 2 feet to 4 feet.

[tex]C(2)=2(2)^2+32=8+32=40[/tex]

[tex]C(4)=2(4)^2+32=32+32=64[/tex]

[tex]\text{Average rate of change=}\frac{C(4)-C(2)}{4-2}[/tex]

[tex]=\frac{64-40}{2}=\frac{24}{2}=$12 per foot[/tex]

Hence, option B is correct.

Answer:

Lengths of AD and AB. They must be same for theorem SAS

Answer:

[tex]\overline{\rm AD} = \overline{\rm AB}[/tex]

Step-by-step explanation:

Two figures are congruent if they have the same shape and size, although their position or orientation are different. The congruence criteria correspond to the postulates and theorems that state what are the minimum conditions that two or more triangles must meet in order to be congruent. One of the congruence criteria is:

SAS (Side-Angle-Side): Two triangles are congruent if they have two sides and the angle determined by them respectively equal.

So, considering the previous information and the data provided by the problem. Then, the additional information necessary to show that triangle DAC is congruent to triangle BAC is:

[tex]\overline{\rm AD} = \overline{\rm AB}[/tex]

The similarity is ΔRST ~ Δ UVW and Scale Factor is 5/6.

If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable. Similar figures are described as items with the same shape but varying sizes, such as two or more figures.

Given:

From the Two Triangles we can see that

<VUW = <SRT = 32

and,   SR / VU =  TR / WU

10/ 12 = 15/ 16

5/6 = 5/6

So, By SAS similarity Criteria both Triangles are Similar.

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Answer:

  A.  slope = -2; point = (8, -3)

Step-by-step explanation:

Compare to the point-slope form for slope m and point (h, k).

  y -k = m(x -h)

You see that k = -3, m = -2, h = 8, so ...