reinout-g

I made a sketch to illustrate your problem. 

I divided the question into 4 sub-questions which should lead to your answer.

Try to solve these step by step and see if this solves your problem.

1. How can we use the equation \(tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \) to find height a and b ? (have a look at https://www.mathsisfun.com/sine-cosine-tangent.html if you don't know how this works)

2. How can we use height a and b to calculate the increase in height of the balloon?

3. How much time has passed for the balloon to cover this distance?

4. If we know how much time has passed and how much distance the balloon has covered, how do we calculate the speed?

Good luck!

All right, so it might be a little too easy for children to get answers to their homework here. However I do hope it's also apparent that there are children here that actually find this forum useful and learn something from our answers? We could indeed improve by giving methods instead of answers however I think this is not very helpful as the poster will still not know if he/she can apply the method correctly. Surely he/she could upload his/her own answers to his website, but this will probably be another threshold as they might feel that still doing it wrong would be considered 'dumb'. Therefore I think Melody provides an excellent solution here by applying the method to a similar problem. 

Hi there,

there's lot's of answers to your question, but I think you're just looking for an example of a fraction that is in between these two.

Let me give you some help here. Suppose I have 1 pizza and I cut it in 4 equal slices then if I take 1 slice I'll have 1/4 pizza. However if I cut the pizza in 8 slices and take 2 slices I'll still have 1/4 pizza.

Now suppose I have 1 pizza and I cut it in 11 equal slices then if I take 1 slice I'll have 1/11 pizza. However if I cut the pizza in 22 slices and take 2 I'll still have 1/11 pizza. Therefore we can say 1/11 = 2/22.

With similar reasoning we can also say that 1/12 = 2/24 (try to think of this pizza experiment yourself.)

So the question 1/11<...<1/12 is actually the same question as  2/22<...<2/24.

Now, I can divide a pizza in 22 slices, I can divide a pizza in 24 slices, but I could also divide it in 23 slices (Don't ask me to actually do it though, it would be very cumbersome).

Now if I have 2 slices of the pizza that is cut in 22 pieces those pieces will be bigger than 2 slices of the pizza that is cut in 23 pieces and those will again be bigger than the 2 slices of the pizza that is cut in 24 pieces.

Therefore we can state 2/22<2/23<2/24.

Now remember that we found that 1/11=2/22 and 1/12=2/24.

Hence we can also write 1/11<2/23<1/12.

And there you have an answer to your question, note however that it is not the only answer.

Suppose I cut the pizza in 44 slices and take 4 for myself I'd also have 1/11th pizza. 

Similarly as above we can therefore also reason that 4/44<4/45<4/46<4/47<4/48 and therefore that

1/11<4/45<4/46<4/47<1/12 (try to follow the reasoning above and do this for yourself.)

I hope this helped 

Reinout

Practice a lot, figure out why something is true instead of simply learning a method or 'trick' and most importantly, don't give up if you're stuck! There are lot's of people to help you here and it's way more rewarding to figure out the answer than to give up and start doing something else.

Don't lose courage, one day you'll find math is not as hard as it look!

Reinout

Let's first take a look at what happens in the first years.

Emily starts out with a salary of $25,000

After one year her salary has increased with 6%.

This means her salary is 106% of what it was last year.

Therefore her salary is now

$$(\frac{106}{100})\times\$25,000 = 1.06 \times\$25,000 = \$26,500$$

One year later her salary again increases with 6%

The means her salary is again 106% of what it was last year

Therefore her salary is now 

$$(\frac{106}{100})\times\$26,500 = 1.06 \times \$26,500 = \$28,090$$

Now suppose we had the same situation and we wanted to know what her salary was after 2 years.

Instead of first calculating her salary after one year we could also have immediately calculated

$$(\text{salary after two years}) = 1.06 \times (\text{salary after one year}) = 1.06 \times 1.06 \times (\text{starting salary}) = (1.06)^2 \times(\text{starting salary}) = (1.06)^2 \times \$25,000 = \$28,090$$

Similarly, we could show that for 3 years we have that 

$$(\text{salary after three years}) = 1.06^3 (\text{starting salary}) = 1.06^3 \times \$25,000$$

and that for x years we have that

$$(\text{salary after x years}) = 1.06^x = (\text{starting salary})$$

Now, if we rewrite the exercise as an equation we need to solve

$$1.06^x \times \$25,000 = 2 \times \$25,000$$

dividing both sides by 25000 gives us

$$1.06^x = 2$$

Now, I'm not sure what your level is so I'll give you two options

option 1 (the easy option (for whole years)):

Start with 1 and keep multiplying by 1.06 until the value is bigger or equal to 2.

The number of times you multiplied is the value of x.

To illustrate, this gives

$$\begin{array}{lcl}
1*1.06 = 1.06\\
1*1.06*1.06 = 1.06^2 = 1.1236\\
1*1.06*1.06*1.06 = 1.06^3 = 1.191016\\
(...)\\
1.06^{11} = 1.898298558\\
1.06^{12} = 2.012196472\\
\end{array}$$

Hence after 12 years, her income has doubled.

Option 2 (the advanced option (for partial years)):

This option makes use of logarithms. If you've never heard of logarithms, this method is not for you.

We indicate the natural logarithm as $$ln() = {}^elog()$$

We have that

$$\begin{array}{lcl}
1.06^x = 2\\
e^{ln(1.06^x)} = 2\\
e^{x \times ln(1.06)} = 2\\
x \times ln(1.06) = ln(2)\\
x = \frac{ln(2)}{ln(1.06)} \approx 11.8957
\end{array}$$

 Hence the answer is approximately 11.8957 years

Reinout 

Well

$$5^2 = 25$$

$$6^2 = 36$$

Which means

$$\sqrt{25} = 5$$

$$\sqrt{36} = 6$$

Hence for any number

$$5 < c < 6$$

$$\sqrt{25} < \sqrt{c} < \sqrt{36}$$

Therefore the only correct answer is

$$a.) \sqrt{28}$$

We find that the radius is half the diameter. Therefore if the diameter is 3.1 meters the radius must be 1.55 meters.

$$1 \frac{1}{2} = \frac{3}{2}$$

$$\frac{{(\frac{3}{2})}}{2} = \frac{3}{2 \cdot 2} = \frac{3}{4}$$

or

$$1\frac{1}{2} = 1.5$$

$$\frac{1.5}{2} = 0.75$$

double post (to be deleted)

We can solve it in this way

$$\begin{array}{lcl}
9^x 3^4 = 3^{10}\\
(3^2)^x 3^4 = 3^{10}\\
3^{2x} 3^4 = 3^{10}\\
3^{2x+4} = 3^{10}\\
2x+4 = 10\\
2x = 6\\
x = 3\\
\end{array}$$

$$\sqrt{25} = 5 \text{ since } 5 \cdot 5 = 25$$

$$(3\sqrt{3})^2 = 3^2 * \sqrt{3}^2 = 9 * 3 = 27$$

I'm not sure whether wood can chuck, but I hope you wooed chuck into chucking wood

Note that there is no single solution for this question since there are 2 variables and only 1 equation. However we do find that the equation is given by

$$5q - 9 = r$$

or similarly

$$q = \frac{r+9}{5}$$

Reinout

First we solve the equation for y

$$\begin{array}{lcl}
x - y = x + y + 17\\
-y = y + 17\\
-2y = 17\\
y = -\frac{17}{2}
\end{array}$$

We have 2 possible situations for x

either 

$$\begin{array}{lcl}
y = -\frac{17}{2}\\
\end{array}$$

and we find

$$\begin{array}{lcl}
x - y = x + y + 17\\
x + \frac{17}{2} = x - \frac{17}{2} + 17\\
x = x - 17 + 17\\
x = x
\end{array}$$

Which means x can be any number,

or

$$\begin{array}{lcl}
y \neq -\frac{17}{2}\\
\end{array}$$

and we find no value for x for which the equation holds.

Reinout

Here's how I solved it

$$\begin{array}{lcl}
e = 2\frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}\\
f = 3\frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3}\\
4e-2f = 4*\frac{5}{2} - 2*\frac{10}{3}\\
= \frac{20}{2} - \frac{20}{3}\\
= \frac{20}{2} * 1 - \frac{20}{3} * 1\\
= \frac{20}{2} * \frac{3}{3} - \frac{20}{3} * \frac{2}{2}\\
= \frac{60}{6} - \frac{40}{6}\\
= \frac{60-40}{6}\\
= \frac{20}{6}\\
= \frac{10}{3}\\
= 3 \frac{1}{3}
\end{array}$$

Reinout