To find the approximate age of the sample, we need to consider the half-life of the starting material, substance A. We also need to consider the relative amounts of substance A and substance B.

Since we are told that the sample we are looking at contains  substance A and  substance B, we know that the sample must have started out with  of substance A. Furthermore, in order to go from  to , we know that a total of two half-lives must have passed. And since we know that one half-life is equal to , we can conclude that a total of  must have passed.

In this question, we're shown a reaction in which platinum is decaying into some unknown element. We're asked to determine how many neutrons this unknown element will have.

In the reaction shown in the question stem, we can see that platinum has  protons, which is its atomic number. The other number shown, , is the mass number. The mass number represents the total number of neutrons and protons. Since we know the mass number and the atomic number, we can determine the number of neutrons in platinum.

Thus, there are  neutrons in platinum. To determine the number of neutrons in the product, however, we need to look at which type of decay we're dealing with. Since an electron is being produced in the reaction shown, we know that this must be beta-decay. In this type of decay, a neutron within the nucleus of the starting atom (platinum) is broken down into a proton and an electron, which is then emitted. Consequently, the product element should have one less neutron that the starting element, platinum. Thus, we can conclude that there will be  neutrons in the new element.

In this question, we're told that a radioactive compound is decaying, and we're asked to determine the rate constant for the process.

Because radioactive decay reactions are first-order, we can use the first-order reaction equation.

Rearranging this expression, we can isolate the rate constant term, , as follows.

Now, we can plug in the values that were given to us in the question stem to calculate the answer.

In this question, we're told that a radioactive compound has decayed by a certain amount in a given amount of time. We're asked to find the rate constant.

First, we have to recognize that all radioactive decay reactions follow first-order kinetics. Hence, we can use the first-order rate equation to solve this question.

Because we're told that  of the compound has decayed, we know there must be  of it left at the  mark.

Fusion is the process of combining two or more atoms to form a larger atom. To remember this, think of how welders fuse metals together. (Though the term is the same, they aren't actually the same thing; this is just to help you remember.). Fusion is a very energetic reaction that takes place in high-heat, high-pressure environments, like the inside of stars. Fusion releases lots of energy, which is why stars are so energetic.

Fission is the process of splitting a signle atom into multiple atoms. It doesn't normally occur in nature, though some super heavy elements, like plutonium, can be spontaneously fissile, which means they can undergo fission seemingly at random. This is a rare thing for an element to do, which is why it's said that fission doesn't normally occur in nature.

First of all, the intermediate nuclear force isn't a real force.

Gravity is not responsible for this, because on the scale of quantum mechanical phenomena, gravity has negligible effect, and can be disregarded.

The electromagnetic force doesn't hold the nucleus together, and is actually trying to rip it apart, due to the fact that like charges repel and the nucleus is full of like charges (protons). Accordingly, the force that actually is responsible for holding it together must necessarily be significantly more powerful compared to the electromagnetic force to resist the intrinsic repelling the protons have towards each other.

The weak nuclear force operates on leptons and quarks, and is involved in many of the radioactive decays in nuclear physics, such as beta decay, where a proton decays into a neutron, where it was first revealed.

Since the other three valid forces aren't responsible, that leaves the strong nuclear force. It is the strongest of the four fundamental forces, as it prevents protons from flying away from each other due to their proximity and charge. The strong force mediates over the quarks that make up the protons and neutrons.

In this question, we're told that an atom undergoes a series of decays. We're then asked to determine how the atomic number of that atom has changed.

Let's look at the first type of decay, alpha decay. During alpha decay, the atom emits a helium nucleus, which consists of two protons and two neutrons. Thus, for each alpha decay, the atom will lose two protons. So two alpha decays would result in a net loss of four protons.

Next, let's look at positron decay. In this type of decay, a proton is converted into a positron and a neutron. The neutron stays in the atoms's nucleus, while the positron is emitted. Thus, positron decay results in a loss of one proton. Consequently, two positron decays result in a total loss of two protons.

Finally, gamma decay does not cause a change in the atom's atomic number or mass number. Gamma decay simply releases energy.

So, in total, we have four protons lost from alpha decays and two protons lost from positron decays. Thus, there is a total loss of six protons, corresponding to a decrease in the atomic number by six.

Write the thin-lens equation.

 represents the object distance from the lens, subtracting the distance between the student's eyes and the lens.

The image distance is:

Substitute the givens to the equation.

Refractive power is the inverse of focal length in .

Let the object distance be  and the image distance be . The object distance is the distance the flower is from the lens. The image distance is the distance the image is located from the lens. The thin lens equation is

Alternatively you can use

To find the height, the equation for the magnification of an object is given by

it is also given as

The negative sign indicates an image that is upside down. Setting these equal gives

Using object distance, image distance, and focal length formula:

Where  is object distance,  is image distance, and  is focal length. 

Plugging everything in:

Solving for ,